CASE STUDIES OF SWEDISH Lathyrus vernus POPULATION
ORGANISM AND POPULATION
Lathyrus vernus (family Fabaceae) is a long-lived forest herb, native to Europe and large parts of northern Asia. Individuals increase slowly in size and usually flower only after 10-15 years of vegetative growth. Flowering individuals have an average conditional lifespan of 44.3 years (Ehrlén and Lehtilä 2002). Lathyrus vernus lacks organs for vegetative spread and individuals are well delimited (Ehrlén 2002). One or several erect shoots of up to 40 cm height emerge from a subterranean rhizome in March-April. Flowering occurs about four weeks after shoot emergence. Shoot growth is determinate, and the number of flowers is determined in the previous year (Ehrlén and Van Groenendael 2001). Individuals may not produce aboveground structures every year but can remain dormant in one season. Lathyrus vernus is self-compatible but requires visits from bumble-bees to produce seeds. Individuals produce few, large seeds and establishment from seeds is relatively frequent (Ehrlén and Eriksson 1996). The pre-dispersal seed predator Bruchus atomarius often consumes a large fraction of developing seeds, and roe deer (Capreolus capreolus) sometimes consume the shoots (Ehrlén and Münzbergová 2009).
Data for this study were collected from six permanent plots in a population of L. vernus located in a deciduous forest in the Tullgarn area, SE Sweden (58.9496 N, 17.6097 E), during 1988–1991 (Ehrlén 1995). The six plots were relatively similar with regard to soil type, elevation, slope, and canopy cover. Within each plot, all individuals were marked with numbered tags that remained over the study period, and their locations were carefully mapped. New individuals were included in the study in each year. Individuals were recorded at least three times every growth season. At the time of shoot emergence, we recorded whether individuals were alive and produced above-ground shoots, and if shoots had been grazed. During flowering, we recorded flower number and the height and diameter of all shoots. At fruit maturation, we counted the number of intact and damaged seeds. To derive a measure of above-ground size for each individual, we calculated the volume of each shoot as \(\pi × (\frac{1}{2} diameter)^2 × height\), and summed the volumes of all shoots. This measure is closely correlated with the dry mass of aboveground tissues (\(R^2 = 0.924\), \(P < 0.001\), \(n = 50\), log-transformed values; Ehrlén 1995). Size of individuals that had been grazed was estimated based on measures of shoot diameter in grazed shoots, and the relationship between shoot diameter and shoot height in non-grazed individuals. Only individuals with an aboveground volume of more than 230 mm3 flowered and produced fruits during this study. Individuals that lacked aboveground structures in one season but reappeared in the following year were considered dormant. Individuals that lacked aboveground structures in two subsequent seasons were considered dead from the year in which they first lacked aboveground structures. Probabilities of seeds surviving to the next year, and of being present as seedlings or seeds in the soil seed bank, were derived from separate yearly sowing experiments in separate plots adjacent to each subplot (Ehrlén and Eriksson 1996).
ANALYSES WITH LATHYRUS DATA
We will complete three different analyses to illustrate some of the ways in which package lefko3 can be used:
through the estimation of raw MPMs, with the intention of producing matrices similar to those published in Ehrlén (2000);
through the estimation of function-based MPMs using a stage classification different from Ehrlén (2000), developed using the natural logarithm of the size measure used in that study; and
through the construction of a complex integral projection model.
Analysis 1: Raw MPM estimation
Here we will estimate raw matrices similar to and based on the dataset used in Ehrlén (2000). These matrices will not be the same, as the dataset included in this package includes data for more individuals as well as an extra year of data. It also includes differences in classification due to different assumptions regarding transitions to and from vegetative dormancy, which is an unobservable life history stage in which herbaceous perennial plants spend the growing season belowground, without aboveground structures. Particularly, we have changed the classification such that plants must have been observed aboveground after a period of of one or more years without producing aboveground sprouts in order to be considered as having been vegetatively dormant (this has the impact of making the survival probability estimated for vegetative dormancy be equal to 1.0 in our analyses). However, the matrices will be very similar.
The dataset that we have provided is organized in horizontal format, meaning that rows correspond to unique individuals and columns correspond to individual condition in particular observation times (which we refer to as years here, since there was one main census in each year). The original Excel spreadsheet used to keep the dataset has a repeating pattern to these columns, with each year having a similarly arranged group of variables. Package lefko3 includes functions to handle data in horizontal format based on these patterns, as well as functions to handle vertically formatted data (i.e. data for individuals is broken up across rows, where each row is a unique combination of individual and year in time t).
Figure 1. Organization of the Lathyrus dataset, as viewed in Microsoft Excel.
This dataset includes information on 1,119 individuals, so there are 1,119 rows with data (not counting the header). There are 38 columns. The first two columns are variables giving identifying information about each individual, with each individual’s data entirely restricted to one row. This is followed by four sets of nine columns, each named VolumeXX, lnVolXX, FCODEXX, FlowXX, IntactseedXX, Dead19XX, DormantXX, Missing19XX, and SeedlingXX, where XX corresponds to the year of observation and with years organized consecutively. Thus, columns 3-11 refer to year 1988, columns 12-20 refer to year 1989, etc. For lefko3 to handle this dataset correctly, we need to know the exact number of years used, which is 4 years here (includes all years from and including 1988 to 1991), we need the columns to be repeated in the same order for each year, and we need years in consecutive order with no extra columns between them.
First, we clear memory and load the dataset.
Step 1. Life history model development
We will now create a stageframe describing the life history of the species and linking it to the data. A stageframe is a data frame that describes all stages in the life history of the organism, in a way usable by the functions in this package and using stage names and classifications that match the data. It needs to include complete descriptions of all stages that occur in the dataset, with each stage defined uniquely. Since this object can be used for automated classification of individuals, all sizes, reproductive states, and other characteristics defining each stage in the dataset need to be accounted for explicitly. This can be difficult if a few data points exist outside the range of sizes specified in the stageframe, so great care must be taken to include all size values and values of other descriptor variables occurring within the dataset. The final description of each stage occurring in the dataset must not completely overlap with any other stage also found in the dataset, although partial overlap is allowed and expected.
Here, we create a stageframe named lathframe based on the classification used in Ehrlén (2000). We build this by creating vectors of the characteristics describing each stage, with each element always in the same order within the vector. Of particular note are the vectors input as sizes and binhalfwidth in the sf_create function. If sizes are to be binned to define stages, then the values in the former vector correspond to the central values of each bin, while values in the latter vector correspond to one-half of the width of the bin. If size values are not to be binned, then narrow binwidths can be used. For example, in this dataset, vegetatively dormant individuals necessarily have a size of 0, and so we can set the halfbinwidth for this stage to 0.5.
sizevector <- c(0, 100, 13, 127, 3730, 3800, 0)
stagevector <- c("Sd", "Sdl", "VSm", "Sm", "VLa", "Flo", "Dorm")
repvector <- c(0, 0, 0, 0, 0, 1, 0)
obsvector <- c(0, 1, 1, 1, 1, 1, 0)
matvector <- c(0, 0, 1, 1, 1, 1, 1)
immvector <- c(1, 1, 0, 0, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0, 0, 0)
indataset <- c(0, 1, 1, 1, 1, 1, 1)
binvec <- c(0, 100, 11, 103, 3500, 3800, 0.5)
lathframe <- sf_create(sizes = sizevector, stagenames = stagevector, repstatus = repvector,
obsstatus = obsvector, matstatus = matvector, immstatus = immvector,
indataset = indataset, binhalfwidth = binvec, propstatus = propvector)
To be useful to others reading your work, or to yourself in the future, it helps to add text descriptions of the stages. Here, we add some descriptions to this stageframe as comments. You may type lathframe at the prompt afterwards to inspect the result.
lathframe$comments <- c("Dormant seed", "Seedling", "Very small vegetative", "Small vegetative",
"Very large vegetative", "Flowering", "Vegetatively dormant")Care should be taken in assigning sizes to stages, particularly when stages occur where size is effectively 0. In most cases, a size of 0 will mean that the individual is alive but unobservable, such as in the case of vegetative dormancy. However, a size of 0 may have different meanings in other cases, such as when the size metric used for classification is a logarithm of true size, making sizes of 0 and lower possible in observable individuals in the dataset. These situations may impact some methods in this package dealing with the construction of function-based MPMs, such as IPMs, particularly in cases where true 0 sizes also occur. See Analysis 2 for an example of such a situation.
Step 2a. Data reorganization
Once the stageframe is created, the dataset should be reorganized into vertical format. Vertically formatted datasets are structured such that each row corresponds to the state of a single individual in two (if ahistorical) or three (if historical) consecutive times. We use the verticalize3() function to handle this and create a historical vertical dataset, as below.
lathvert <- verticalize3(lathyrus, noyears = 4, firstyear = 1988, patchidcol = "SUBPLOT",
individcol = "GENET", blocksize = 9, juvcol = "Seedling1988",
size1col = "Volume88", repstr1col = "FCODE88", fec1col = "Intactseed88",
dead1col = "Dead1988", nonobs1col = "Dormant1988",
stageassign = lathframe, stagesize = "sizea", censorcol = "Missing1988",
censorkeep = NA, censor = TRUE)
It pays to explore a dataset thoroughly prior to full analysis, and summaries can certainly help to do that. Type summary(lathvert) at the prompt to look at such a summary. This summary will reveal that the verticalize3() function has automatically subset the data to only those instances in which the individual is alive in time t (please take a look at the variable alive2). Knowing this can help us interpret other variables. For example, the mean value for alive3 suggests very high survival to time t+1. Further, the minimum values for all size variables are 0, suggesting that unobservable stages occur within the dataset (as already mentioned, these are instances of vegetative dormancy, which is treated as a separate life history stage in this example).
Subset summaries may shed more light on these data. For example, after subsetting to only cases in which individuals are vegetatively dormant in time t and narrowing the summary to only variables corresponding to time t (columns 22 to 33), we see that this particular stage is associated exclusively with a size of 0. We can also see from the dimensions of this subset that vegetative dormancy occurs often in this dataset.
summary(lathvert[which(lathvert$stage2 == "Dorm"),c(22:33)])
#> sizea2 repstra2 feca2 fec2added juvgiven2 obsstatus2 repstatus2 fecstatus2 matstatus2 alive2
#> Min. :0 Min. : NA Min. : NA Min. :0 Min. :0 Min. :0 Min. :0 Min. :0 Min. :1 Min. :1
#> 1st Qu.:0 1st Qu.: NA 1st Qu.: NA 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:1 1st Qu.:1
#> Median :0 Median : NA Median : NA Median :0 Median :0 Median :0 Median :0 Median :0 Median :1 Median :1
#> Mean :0 Mean :NaN Mean :NaN Mean :0 Mean :0 Mean :0 Mean :0 Mean :0 Mean :1 Mean :1
#> 3rd Qu.:0 3rd Qu.: NA 3rd Qu.: NA 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:1 3rd Qu.:1
#> Max. :0 Max. : NA Max. : NA Max. :0 Max. :0 Max. :0 Max. :0 Max. :0 Max. :1 Max. :1
#> NA's :138 NA's :138
#> stage2 stage2index
#> Length:138 Min. :7
#> Class :character 1st Qu.:7
#> Mode :character Median :7
#> Mean :7
#> 3rd Qu.:7
#> Max. :7
#>
dim(lathvert[which(lathvert$stage2 == "Dorm"),])
#> [1] 138 45
The fact that vegetatively dormant individuals have been assigned 0 for size does not require us to exercise any extra steps, because the construction of raw matrices depends on the stage designations rather than on the size classifications. In this case, the verticalize3() function has made these assignments, and this can be seen in the summary in the stage2index column, which shows all individuals that are alive and have a size of 0 in time t to be in the 7th stage. Type lathframe and enter at the prompt to check that the 7th stage in the stageframe is really vegetative dormancy.
A further subset summary can teach us how reproduction is handled here. The reproductive status of flowering adults is certainly set as reproductive (see the distribution of values for repstatus2). However, fecundity ranges from 0 to the high 60s (see feca2, which codes for fecundity in time t). This has happened because the reproductive status variable that we used , FCODE88, notes whether these individuals flowered but not whether they produced seed. Since this plant is not obligately selfing, and since even self-reproduction requires pollen deposition by an insect vector, many flowers will yield no seed. This issue does not cause problems for us here, but it may cause difficulties in the creation of function-based matrices if we wished to assume a count-based distribution for fecundity. In any case, it helps to consider whether the definitions used for stages are appropriate, and so whether reproductive status must necessarily be associated with successful reproduction or merely the attempt. Here, we associate it with the latter, but in Analyses 2 and 3 we will associate it with the former.
summary(lathvert[which(lathvert$stage2 == "Flo"),c(22:33)])
#> sizea2 repstra2 feca2 fec2added juvgiven2 obsstatus2 repstatus2 fecstatus2 matstatus2
#> Min. : 98.4 Min. :1 Min. : 0.000 Min. : 0.000 Min. :0 Min. :1 Min. :1 Min. :0.0000 Min. :1
#> 1st Qu.: 732.5 1st Qu.:1 1st Qu.: 0.000 1st Qu.: 0.000 1st Qu.:0 1st Qu.:1 1st Qu.:1 1st Qu.:0.0000 1st Qu.:1
#> Median :1141.8 Median :1 Median : 0.000 Median : 0.000 Median :0 Median :1 Median :1 Median :0.0000 Median :1
#> Mean :1388.5 Mean :1 Mean : 4.793 Mean : 4.793 Mean :0 Mean :1 Mean :1 Mean :0.4424 Mean :1
#> 3rd Qu.:1758.0 3rd Qu.:1 3rd Qu.: 6.000 3rd Qu.: 6.000 3rd Qu.:0 3rd Qu.:1 3rd Qu.:1 3rd Qu.:1.0000 3rd Qu.:1
#> Max. :7032.0 Max. :1 Max. :66.000 Max. :66.000 Max. :0 Max. :1 Max. :1 Max. :1.0000 Max. :1
#> alive2 stage2 stage2index
#> Min. :1 Length:599 Min. :6
#> 1st Qu.:1 Class :character 1st Qu.:6
#> Median :1 Mode :character Median :6
#> Mean :1 Mean :6
#> 3rd Qu.:1 3rd Qu.:6
#> Max. :1 Max. :6Voilá! Now we will move on to creating a few extra objects that will help us estimate full population projection matrices.
Step 2b. Provide supplemental information for matrix estimation
Now we will create a reproductive matrix. This matrix shows which stages are reproductive, which stages they lead to the production of, and at what level reproduction occurs. This matrix is mostly composed of 0s, but fecundity is noted as non-zero entries equal to a scalar multiplier to the full fecundity estimated by lefko3. This matrix has rows and columns equal to the number of stages described in the stageframe for this dataset, and the rows and columns refer to these stages in the same order as in the stageframe. In many ways, this matrix looks like a nearly empty ahistorical population matrix, but notes the per-individual mean multipliers on fecundity for each stage that actually reproduces.
First, we will create a 0 matrix with dimensions equal to the number of rows in lathframe. Then we will modify elements corresponding to fecundity by the expected mean seed dormancy probability (row 1), and by the germination rate (row 2). Since only stage 6 (flowering adult, or Flo) is reproductive, and since reproduction leads to the production of individuals in stage 1 (dormant seed, or Sd) and stage 2 (seedlings, or Sdl), we will alter matrix elements [1, 6] and [2, 6].
lathrepm <- matrix(0, 7, 7)
lathrepm[1, 6] <- 0.345
lathrepm[2, 6] <- 0.054
lathrepm
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0 0 0 0 0 0.345 0
#> [2,] 0 0 0 0 0 0.054 0
#> [3,] 0 0 0 0 0 0.000 0
#> [4,] 0 0 0 0 0 0.000 0
#> [5,] 0 0 0 0 0 0.000 0
#> [6,] 0 0 0 0 0 0.000 0
#> [7,] 0 0 0 0 0 0.000 0Next we will provide some given transitions via an overwrite table. Specifically, we will provide the seed dormancy probability and germination rate, which are given as transitions from the dormant seed stage to another year of seed dormancy or to the germinated seedling stage, respectively. We assume that the germination rate is the same regardless of whether the seed was produced in the previous year or has been in the seedbank for longer. We will start with the ahistorical case.
lathover2 <- overwrite(stage3 = c("Sd", "Sdl"), stage2 = c("Sd", "Sd"),
givenrate = c(0.345, 0.054))
lathover2
#> stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate convtype
#> 1 Sd Sd NA NA NA NA 0.345 1
#> 2 Sdl Sd NA NA NA NA 0.054 1
And now the historical case, where we also need to input the corresponding stages in time t-1 for each transition. Going back a further step to look at time t-1 shows us that there are two ways that a dormant seed in time t can be a dormant seed in time t+1 - it could have been a dormant seed in time t-1, or it could have been produced by a reproductive individual in time t-1. Thus, we will enter 3 given transitions in the historical case, rather than 2 transitions, as in the ahistorical case. Note the use of the "rep" designation for Stage1 - this is shorthand telling R to use all reproductive stages for the time interval. Here, there is only one reproductive stage, but in other cases, such as in an IPM, this shorthand notation can save time and hassle by representing many stages in a single statement.
lathover3 <- overwrite(stage3 = c("Sd", "Sd", "Sdl"), stage2 = c("Sd", "Sd", "Sd"),
stage1 = c("Sd", "rep", "rep"), givenrate = c(0.345, 0.345, 0.054))
lathover3
#> stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate convtype
#> 1 Sd Sd Sd NA NA NA 0.345 1
#> 2 Sd Sd rep NA NA NA 0.345 1
#> 3 Sdl Sd rep NA NA NA 0.054 1
These two overwrite tables show us that we have survival-transition probabilities (convtype = 1, whereas fecundity rates would be given as convtype = 2), that the given transitions originate from the dormant seed stage (Sd) in time t (and seeds or reproductive stages in time t-1 in the historical case), and the specific values to be used in overwriting: 0.345 and 0.054. If we wished, we could have used the values of transitions to be estimated within this matrix as proxies for these values, in which case the eststageX columns would have entries corresponding to the stages of the correct transitions and the givenrate column would be blank (see the Cypripedium candidum vignette for examples).
Now we are ready to create some ahistorical and historical MPMs!
Step 3. Tests of history
For comparison purposes, we have chosen to build both ahistorical and historical MPMs in this vignette. However, in a typical analysis, it is most parsimonious to test whether history influences the demography of the population significantly first, and only use historical MPMs if the test supports the hypothesis that it does. A number of methods exist to conduct these tests, and we recommend Brownie et al. (1993), Pradel (2003), Pradel et al. (2005), and Cole et al. (2014) for good discussions and tools to help with this.
In lefko3, we provide a function that can be used to test history directly from the historical vertical dataset: modelsearch(). This function is described in detail in the Basic theory and concepts vignette and in Analyses 2 and 3, in which it is used to create function-based MPMs. We provide only a barebones description here, focusing only on how to test for the effects of history on demography, and encourage readers to read the more detailed descriptions noted above to get a better understanding of this function.
Function modelsearch() estimates best-fit linear models of the key vital rates used to propagate elements in function-based MPMs. There are up to 9 different vital rates possible to test, of which 5 are adult vital rates and 4 are juvenile vital rates. The standard vital rates that we may wish to test are survival (marked as surv in vitalrates), size (size), and fecundity (fec). Here, we also test observation status (obs), which can serve as a proxy for sprouting probability in cases where plants do not necessarily sprout. To test history with function modelsearch, use the historical vertical dataset as input, set historical = TRUE to make sure that linear model, input the relevant vital rates to estimate, set the suite of independent factors to test (size and reproductive status in times t and t-1 and all interactions, or some subset thereof), set the name of the juvenile stage (if juvenile vital rates are to be estimated), set the proper distributions to use for size and fecundity, and note which variables code for individual identity (used to treat identity as a random factor in mixed linear models) and observation time. We also use quiet = TRUE to limit the amount of text output while the function runs.
histtest <- modelsearch(lathvert, historical = TRUE, suite = "size", vitalrates = c("surv",
"obs", "size", "fec"), juvestimate = "Sdl", sizedist = "gaussian",
fecdist = "gaussian", indiv = "individ", year = "year2",
quiet = TRUE)
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0189452 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: very large eigenvalue
#> - Rescale variables?;Model is nearly unidentifiable: large eigenvalue ratio
#> - Rescale variables?
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
histtest[1:9]
#> $survival_model
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea1 + (1 | individ) + (1 | year2)
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 910.3733 933.2409 -451.1866 902.3733 2242
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.2571
#> year2 (Intercept) 0.6149
#> Number of obs: 2246, groups: individ, 257; year2, 3
#> Fixed Effects:
#> (Intercept) sizea1
#> 2.589549 0.001786
#> convergence code 0; 0 optimizer warnings; 3 lme4 warnings
#>
#> $observation_model
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: obs.data
#> AIC BIC logLik deviance df.resid
#> 1353.5346 1370.5136 -673.7673 1347.5346 2118
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.01967
#> year2 (Intercept) 0.00000
#> Number of obs: 2121, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.235
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> $size_model
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ sizea1 + sizea2 + (1 | individ) + (1 | year2) + sizea1:sizea2
#> Data: size.data
#> REML criterion at convergence: 29132.25
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.0
#> year2 (Intercept) 247.2
#> Residual 480.4
#> Number of obs: 1916, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept) sizea1 sizea2 sizea1:sizea2
#> 8.998e+01 3.119e-01 5.954e-01 -9.417e-05
#> fit warnings:
#> Some predictor variables are on very different scales: consider rescaling
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> $repstatus_model
#> [1] 1
#>
#> $fecundity_model
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: feca2 ~ sizea2 + (1 | individ) + (1 | year2)
#> Data: fec.data
#> REML criterion at convergence: 4147.316
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 2.187
#> year2 (Intercept) 2.461
#> Residual 7.286
#> Number of obs: 600, groups: individ, 128; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 0.829147 0.002794
#>
#> $juv_survival_model
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ (1 | individ) + (1 | year2)
#> Data: juvsurv.data
#> AIC BIC logLik deviance df.resid
#> 323.6696 334.5847 -158.8348 317.6696 278
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0
#> year2 (Intercept) 0
#> Number of obs: 281, groups: individ, 187; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 1.084
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> $juv_observation_model
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: juvobs.data
#> AIC BIC logLik deviance df.resid
#> 91.4924 101.5338 -42.7462 85.4924 207
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 16.009
#> year2 (Intercept) 1.221
#> Number of obs: 210, groups: individ, 154; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 10.39
#>
#> $juv_size_model
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ (1 | individ) + (1 | year2)
#> Data: juvsize.data
#> REML criterion at convergence: 1303.003
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.989
#> year2 (Intercept) 1.182
#> Residual 6.997
#> Number of obs: 193, groups: individ, 144; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 11.02
#>
#> $juv_reproduction_model
#> [1] 1
The object generated is quite long, but for our purposes we only need to look at elements corresponding to the best-fit models for our tested vital rates, which are the 9 elements marked $survival_model, $observation_model, $size_model, $repstatus_model, $fecundity_model, $juv_survival_model, $juv_observation _model, $juv_size_model, and $juv_reproduction_model (we have excluded the rest of the object from being printed with the [1:9] tag). The line beginning Formula: in each of these sections shows the best-fit model, in standard R formula notation (i.e. \(y = ax + b\) is given as y ~ x). The independent terms tested include variables coding for size in times t and t-1 (in this case, given as sizea2 and sizea1, respectively). We did not test for the impacts of reproductive status in those time because there is only one reproductive stage, and so the output for that model above is given as 1, which simply denotes that all individuals marked as being in a reproductive stage in the dataset are assumed to be reproductive. However, it is possible to test for the impacts of this factor as well.
Since several vital rates show sizea1 as a term in the best-fit model, particularly adult survival and size, we see that history has a significant impact on the demography of this population and cannot be ignored.
We will now move on to MPM estimation.
Step 4. MPM estimation
Now let’s create some raw Lefkovitch MPMs based on Ehrlén (2000). We have seen that history should be included in these analyses, which justifies creating only historical matrices. However, to introduce these functions in greater depth and detail, we will also create the ahistorical MPMs, and will begin with the latter.
Ehrlén (2000) shows a mean matrix covering years 1989 and 1990 as time t. We will utilize the entire dataset instead, covering 1988 to 1991, as follows.
ehrlen2 <- rlefko2(data = lathvert, stageframe = lathframe, year = "all",
stages = c("stage3", "stage2"), repmatrix = lathrepm,
overwrite = lathover2, yearcol = "year2", indivcol = "individ")
The output from this analysis is a lefkoMat object, which is an S3 object (list) with the following elements:
A: a list of full population projection matrices, in order of population, patch, and year (order given in $labels)
U: a list of matrices where non-zero entries are limited to survival-transition elements, in the same order as A
F: a list of matrices where non-zero entries are limited to fecundity elements, in the same order as A
hstages: a data frame showing the order of paired stages (given if matrices are historical, otherwise NA)
ahstages: the stageframe used in analysis, with stages potentially reordered and edited as they occur in the matrix
labels: a table showing the order of matrices by population, patch, and year
matrixqc: a short vector used in summary statements to describe the overall quality of each matrix (used in summary() calls)
dataqc: a short vector used in summary statements to describe key sampling aspects of the dataset (used in summary() calls)
Type ehrlen2 to look at the output in more detail (we have declined to do so here because of the size of the output).
The input for the rlefko2() function includes year = "all", which can be changed to year = c(1989, 1990) to focus just on years 1989 and 1990, as in the paper, or year = 1989 to focus exclusively on the transition from 1989 to 1990 (the year entered is the year in time t). Matrix-estimating functions in lefko3 have a default behavior of creating a matrix for each year in the dataset, rather than lumping years. However, patches will only be separated if a patch ID variable is provided as input (we did not do so here). Package lefko3 includes a great deal of flexibility here, and can quickly estimate many matrices covering all of the populations, patches, and years occurring in a specific dataset. The function-based matrix approach in the next section will showcase more of this flexibility.
We can understand lefkoMat objects in greater detail through the summary function.
summary(ehrlen2)
#>
#> This ahistorical lefkoMat object contains 3 matrices.
#>
#> Each matrix is a square matrix with 7 rows and columns, and a total of 49 elements.
#> A total of 74 survival transitions were estimated, with 24.6666666666667 per matrix.
#> A total of 6 fecundity transitions were estimated, with 2 per matrix.
#>
#> The dataset contains a total of 276 unique individuals and 2527 unique transitions.
#> NULLWe start off learning that 3 full A matrices were estimated (as well as their U and F decompositions), and that they were ahistorical. This is expected given that there are 4 consecutive years of data, yielding 3 time steps, and an ahistorical matrix requires two consecutive years to estimate transitions. The following line notes the dimensions of those matrices. The third, fourth, and fifth lines of the summary are particularly important: they show how many survival transition and fecundity elements were actually estimated, both overall and per matrix, and the number of individuals and transitions the matrices are based on.
These numbers can be used to understand the matrices in greater depth, particularly in two ways. First, the matrices generated might be overparameterized, meaning that the number of elements estimated is too high given the size of the dataset, and this summary can give a sense of whether this is a possibility. Second, they provide an idea of the overall level of statistical power and the potential for pseudoreplication. It is typical for population ecologists to consider the total number of transitions in a dataset as an indication of statistical power when creating ahMPMs. However, the number of individuals used is just as important because each transition that an individual experiences is dependent on the other transitions that it also experiences. Indeed, this is the fundamental point that led to the development of historical matrices and of this package - the assumption that the status of an individual in the next time is dependent only on its current state is too simplistic and leads to pseudoreplication, unincorporated trade-offs, and perhaps other problems. While classic pseudoreplication should not be an issue in raw matrices, it is very likely in function-based MPMs if nothing is attempted to correct for the inclusion of multiple data points from the same individual. Although we do not offer rules of thumb for determining whether the numbers of individuals, transitions, and estimated matrix elements are sufficient for any particular analysis, we believe that the transparency offered in this summary can help users explore these issues on their own.
Now let’s create the historical matrices. Because of the size of these matrices, we will only show the summary.
ehrlen3 <- rlefko3(data = lathvert, stageframe = lathframe, year = "all",
stages = c("stage3", "stage2", "stage1"), repmatrix = lathrepm,
overwrite = lathover3, yearcol = "year2", indivcol = "individ")
summary(ehrlen3)
#>
#> This historical lefkoMat object contains 2 matrices.
#>
#> Each matrix is a square matrix with 49 rows and columns, and a total of 2401 elements.
#> A total of 151 survival transitions were estimated, with 75.5 per matrix.
#> A total of 14 fecundity transitions were estimated, with 7 per matrix.
#>
#> The dataset contains a total of 276 unique individuals and 2527 unique transitions.
#> NULLThe summary output shows a number of fundamental differences. First, there is one less matrix estimated in the historical case than in the ahistorical case because raw historical matrices require three consecutive times to estimate each transition instead of two. Second, these matrices are quite a bit bigger than ahistorical matrices, with the number of rows and columns generally equaling the number of ahistorical rows and columns squared (although this number may sometimes be reduced). Finally, a much greater proportion of each matrix is composed of 0s in the historical case than in the ahistorical case, although there are certainly more non-zero elements. This is because historical matrices are primarily composed of structural 0s. The result is that the historical matrices in this example have non-zero entries in 82.5 / 2401 = 3.4% of matrix elements, while the equivalent ahistorical matrices have non-zero entries in 26.667 / 49 = 54.45% of matrix elements.
We can see the impact of structural 0s by eliminating some of them in the process of matrix estimation. The easy way to do so is to set reduce = TRUE within the rlefko3() call. This will eliminate stage pairs in which both column and row are zero vectors. This will end up giving us matrices with 19 fewer rows and columns, and with 82.5 / 900 = 9.2% of the resulting matrix elements estimated.
ehrlen3red <- rlefko3(data = lathvert, stageframe = lathframe, year = c(1989, 1990),
stages = c("stage3", "stage2", "stage1"), repmatrix = lathrepm,
overwrite = lathover3, yearcol = "year2",
indivcol = "individ", reduce = TRUE)
summary(ehrlen3red)
#>
#> This historical lefkoMat object contains 2 matrices.
#>
#> Each matrix is a square matrix with 30 rows and columns, and a total of 900 elements.
#> A total of 151 survival transitions were estimated, with 75.5 per matrix.
#> A total of 14 fecundity transitions were estimated, with 7 per matrix.
#>
#> The dataset contains a total of 276 unique individuals and 2527 unique transitions.
#> NULL
Next we will create a mean ahistorical matrix.
ehrlen2mean <- lmean(ehrlen2)
print(ehrlen2mean$A, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.345 0.0000 0.00000 0.0000 0.00000 1.5786 0.0000
#> [2,] 0.054 0.0000 0.00000 0.0000 0.00000 0.2471 0.0000
#> [3,] 0.000 0.6637 0.72405 0.0961 0.00735 0.0000 0.0810
#> [4,] 0.000 0.0189 0.12249 0.5804 0.19527 0.1003 0.2664
#> [5,] 0.000 0.0000 0.00131 0.0773 0.31232 0.2327 0.1550
#> [6,] 0.000 0.0000 0.00000 0.0735 0.34499 0.5719 0.0616
#> [7,] 0.000 0.0628 0.02867 0.1096 0.10605 0.0858 0.1026
Function lmean() creates a lefkoMat object, just as rlefko2() and rlefko3() do. So, the output includes the main composite mean matrices (shown in element A), as well as the mean survival-transition matrix (U) and the mean fecundity matrix (F), followed by a section outlining the definitions and order of historical paired stages (hstages, shown here as NA because the matrices are ahistorical), a section outlining the actual stages as outlined in the stageframe object used to create these matrices (ahstages), a section outlining the definitions and order of the matrices (labels), and then two quality control sections used in output for the summary() function (matrixqc and dataqc). So, all descriptive information from the original lefkoMat objects is retained.
Users will note in the output that we have a single mean matrix. The default setting for lmean() outputs both patch and population means. However, here we did not separate patches in the dataset, so the patch mean is equal to the population mean. In that situation, lmean() outputs a single mean matrix.
To see the exact specifications for each matrix, we can look at the labels component of our mean lefkoMat object.
The 1 designation is the default symbol used for population and patch when no designations are supplied. This scenario is typical when no such division is made in the dataset, or no such division is provided to the matrix-estimating functions generating the MPM.
We may wish to check for errors by assessing the survival of ahistorical stages. The following shows us stage survival as the column sums for the main survival-transition matrix.
All of these values are within the realm of possibility for probabilities. If we look back at the stageframe for this analysis to get the order of stages, we find that they are also reasonably similar to the values published in Ehrlén (2000). So this matrix appears to be fine at first glance. Additional approaches to error-checking can include checks of specific elements within the matrix to see if they match the expected values, and we certainly encourage this approach because it produces a familiarity with both the matrices produced as well as with the peculiarities of the functions used to generate them. This is the fine-scale approach that we, the maintainers of package lefko3, take to isolate and fix errors in the code underlying these functions.
Now we will estimate the historical mean matrix. We will use the reduced matrix to simplify later analyses, and show only the top-left corner of the rather large matrix.
ehrlen3mean <- lmean(ehrlen3red)
print(ehrlen3mean$A[[1]][1:20,1:8], digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 0.345 0 0.0000 0.000 0 0.00000 0.000 0
#> [2,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [3,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [4,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [5,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [6,] 0.000 0 0.6880 0.000 0 0.79374 0.000 0
#> [7,] 0.000 0 0.0330 0.000 0 0.07556 0.000 0
#> [8,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [9,] 0.000 0 0.0269 0.000 0 0.00505 0.000 0
#> [10,] 0.000 0 0.0000 0.333 0 0.00000 0.542 0
#> [11,] 0.000 0 0.0000 0.417 0 0.00000 0.458 0
#> [12,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [13,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [14,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [15,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [16,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [17,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [18,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [19,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0
#> [20,] 0.000 0 0.0000 0.000 0 0.00000 0.000 0The prevalence of 0s in this matrix is normal because most elements are structural 0s and so cannot equal anything else. This matrix is also a raw matrix, meaning that transitions that do not actually occur in the dataset cannot equal anything other than 0.
To understand the dominance of structural 0s in the historical case, let’s take a look at the hstages object associated with this mean matrix.
ehrlen3mean$hstages
#> stage_id_2 stage_id_1 stage_2 stage_1
#> 1 1 1 Sd Sd
#> 2 2 1 Sdl Sd
#> 3 3 2 VSm Sdl
#> 4 4 2 Sm Sdl
#> 5 7 2 Dorm Sdl
#> 6 3 3 VSm VSm
#> 7 4 3 Sm VSm
#> 8 5 3 VLa VSm
#> 9 7 3 Dorm VSm
#> 10 3 4 VSm Sm
#> 11 4 4 Sm Sm
#> 12 5 4 VLa Sm
#> 13 6 4 Flo Sm
#> 14 7 4 Dorm Sm
#> 15 3 5 VSm VLa
#> 16 4 5 Sm VLa
#> 17 5 5 VLa VLa
#> 18 6 5 Flo VLa
#> 19 7 5 Dorm VLa
#> 20 1 6 Sd Flo
#> 21 2 6 Sdl Flo
#> 22 4 6 Sm Flo
#> 23 5 6 VLa Flo
#> 24 6 6 Flo Flo
#> 25 7 6 Dorm Flo
#> 26 3 7 VSm Dorm
#> 27 4 7 Sm Dorm
#> 28 5 7 VLa Dorm
#> 29 6 7 Flo Dorm
#> 30 7 7 Dorm Dorm
There are 30 pairs of ahistorical stages. These pairs correspond to the stages that the rows and columns of the historical matrices output by rlefko3() refer back to. The pairs are interpreted so that matrix columns represent the stages in times t-1 and t, and the rows represent stages in times t and t+1. For an element in the matrix to contain a number other than 0, it must first represent the same stage at time t in both the column stage pairs and the row stage pairs. The element [1, 1], for example, represents the transition probability from dormant seed at times t-1 and t (column pair), to dormant seed at times t and t+1 (row pair) - the time t stages match, and so this element is possible. However, element [1, 2] represents the transition probability from seedling in time t-1 and very small adult in time t (column pair), to dormant seed in time t and in time t+1 (row pair). Clearly [1, 2] is a structural 0 because it is impossible for an individual to be both a dormant seed and a very small adult at the same time.
Error-checking is more difficult with historical matrices because they are typically one or two orders of magnitude bigger than their ahistorical counterparts, but the same basic strategy can be used here as with ahistorical matrices. In these cases we can use summary() to assess the distribution of survival estimates for historical stages, which is given as the set of column sums in the survival-transition matrix, as below.
print(summary(colSums(ehrlen3mean$U[[1]])), digits = 3)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.000 0.500 0.922 0.724 1.000 1.000
As long as all of the numbers are between 0 and 1, then all is well AT FIRST GLANCE. Fine-scale error-checking is of great help for historical matrices, and requires outputting the matrix into a spreadsheet and assessing the locations and values of non-zero elements using the hstages output as a guide to what the elements refer to. Non-zero elements should only exist where biologically and logically possible, and in the case of raw matrices, where individuals were actually recorded transitioning.
Step 5. MPM analysis
Package lefko3 includes functions to conduct some analyses of population dynamics. Some of the first analyses users may wish to conduct are deterministic analyses, assessing \(\lambda\), the stable stage distribution, and the reproductive value vector for estimated matrices. We will start by estimating the asymptotic population growth rate (\(\lambda\)) from the ahistorical MPMs, followed by the population growth rate associated with the mean matrix from that analysis. Note that each \(\lambda\) estimate also includes a data frame describing the matrices in order (this is the labels object within the output list). Here is the set of ahistorical annual \(\lambda\) estimates.
lambda3(ehrlen2)
#> pop patch year2 lambda
#> 1 NA NA 1988 0.8952585
#> 2 NA NA 1989 0.9235493
#> 3 NA NA 1990 1.0096490Here is the \(\lambda\) value associated with the mean matrix.
Readers may be surprised to see that \(\lambda\) for the mean matrix is on the high side relative to the annual matrices. However, elements in raw annual matrices may differ dramatically due simply to a lack of individuals transitioning through them in a particular year, which is a situation more likely to happen with smaller datasets and larger numbers of recognized stages. Since different elements within the matrix have different amounts of influence on the population growth rate itself, \(\lambda\) may differ in ways difficult to predict without conducting a perturbation analysis.
We will now look at the same numbers for the historical analyses. First the annual matrices.
And now the mean matrix.
Readers will likely observe both that there are fewer \(\lambda\) estimates in the historical case, and that mean \(\lambda\) is lower. First, because there are 4 years of data, there are three ahistorical transitions possible for estimation: year 1 to 2, year 2 to 3, and year 3 to 4. However, in the historical case, only two are possible: from years 1 and 2 to 3 (technically, from years 1 [t-1] and 2 [t] to years 2 [t] and 3 [t+1]), and from years 2 and 3 to 4 (technically, from years 2 [t-1] and 3 [t] to years 3 [t] and 4 [t+1]). Second, historical matrices cover more of the individual heterogeneity in a population by splitting ahistorical transitions by stage in time t-1. This heterogeneity may reflect the impacts of trade-offs operating across years (Shefferson and Roach 2010). One particularly common trade-off is the cost of growth: an individual that grows a great deal in one time step due to great environmental conditions in that year might pay a large cost of survival, growth, or reproduction in the next if those environmental conditions deteriorate (Shefferson, Warren, and Pulliam 2014; Shefferson et al. 2018). Alternatively, these patterns may reflect greater year-to-year variability in allocation to aboveground tissues relative to belowground tissues, the latter of which includes this species’ perennating structure. While we do not argue that the drop in \(\lambda\) must be due to these issues, and it is certainly possible for historical matrix analysis to lead to higher \(\lambda\) estimates, we do believe that this \(\lambda\) is likely to be more realistic than the higher \(\lambda\) estimated in the ahistorical case.
We can also examine the stable stage distributions, as follows for the ahistorical case. Our numbers here match those produced by package popbio’s stable.stage() function.
ehrlen2mss <- stablestage3(ehrlen2mean)
ehrlen2mss
#> matrix stage_id stage ss_prop
#> 1 1 1 Sd 0.29261073
#> 2 1 2 Sdl 0.04579994
#> 3 1 3 VSm 0.22875637
#> 4 1 4 Sm 0.18625613
#> 5 1 5 VLa 0.07716891
#> 6 1 6 Flo 0.11352077
#> 7 1 7 Dorm 0.05588715
The data frame output shows us the stages themselves (stage, and associated number in stage_id), which matrix they refer to (matrix), and the stable stage distribution (ss_prop). Interpreting these values, we find that the mean matrix suggests that, if we project the population forward indefinitely assuming the population dynamics are static and represented by this matrix, we will find that approximately 29% of individuals should be dormant seeds (suggesting a large seedbank), a further 23% and 19% should be very small and small adults, respectively, and 11% should be flowering adults. Almost 7% of the population should eventually be composed of vegetatively dormant adults.
We can estimate the stable stage distribution for the historical case, as well. Because the historical output for the stablestage3() function is a list with two data frames, let’s take a look at each of these data frames in turn. The first will be the stage-pair output.
ehrlen3mss <- stablestage3(ehrlen3mean)
ehrlen3mss$hist
#> matrix stage_id_2 stage_id_1 stage_2 stage_1 ss_prop
#> 1 1 1 1 Sd Sd 0.1276315506
#> 2 1 2 1 Sdl Sd 0.0123574691
#> 3 1 3 2 VSm Sdl 0.0000000000
#> 4 1 4 2 Sm Sdl 0.0000000000
#> 5 1 7 2 Dorm Sdl 0.0000000000
#> 6 1 3 3 VSm VSm 0.1461199792
#> 7 1 4 3 Sm VSm 0.0241582477
#> 8 1 5 3 VLa VSm 0.0000410067
#> 9 1 7 3 Dorm VSm 0.0018685873
#> 10 1 3 4 VSm Sm 0.0285657384
#> 11 1 4 4 Sm Sm 0.1119221216
#> 12 1 5 4 VLa Sm 0.0164336036
#> 13 1 6 4 Flo Sm 0.0151800785
#> 14 1 7 4 Dorm Sm 0.0204484551
#> 15 1 3 5 VSm VLa 0.0002225478
#> 16 1 4 5 Sm VLa 0.0169172806
#> 17 1 5 5 VLa VLa 0.0249610073
#> 18 1 6 5 Flo VLa 0.0290060920
#> 19 1 7 5 Dorm VLa 0.0100711883
#> 20 1 1 6 Sd Flo 0.2069917052
#> 21 1 2 6 Sdl Flo 0.0323987017
#> 22 1 4 6 Sm Flo 0.0172203233
#> 23 1 5 6 VLa Flo 0.0285095637
#> 24 1 6 6 Flo Flo 0.0663126671
#> 25 1 7 6 Dorm Flo 0.0108687116
#> 26 1 3 7 VSm Dorm 0.0040092199
#> 27 1 4 7 Sm Dorm 0.0251695448
#> 28 1 5 7 VLa Dorm 0.0112789851
#> 29 1 6 7 Flo Dorm 0.0041534402
#> 30 1 7 7 Dorm Dorm 0.0071821838
This data frame is structured in historical format, and so shows the stable stage distribution of stage pairs. We may wish to see which stage pair dominates, in which case we might look at the row with the maximum ss_prop value.
ehrlen3mss$hist[which(ehrlen3mss$hist$ss_prop == max(ehrlen3mss$hist$ss_prop)),]
#> matrix stage_id_2 stage_id_1 stage_2 stage_1 ss_prop
#> 20 1 1 6 Sd Flo 0.2069917Here we see that about 21% of the population is expected to be composed of dormant seeds just produced in the preceding year. This provides an added nuance to the ahistorical results, since dormant seeds could also have been dormant in the preceding year. However, the population is expected to be composed of 13% dormant seeds that were previously dormant, suggesting that new dormant seeds should be more common. This very likely reflects the mortality of dormant seeds, which is assumed to be 65.5% annually.
The longer format of the historical stable stage output makes it a bit hard to read. However, these historical values can also be combined by stage at time t (stage_2) to estimate the historically-corrected stable stage distribution, which allows comparison to a stable stage distribution estimated from a purely ahistorical MPM. We can do this by examining the $ahist element of the historical stable stage distribution object.
ehrlen3mss$ahist
#> matrix stage_id stage ss_prop
#> 1 1 1 Sd 0.33462326
#> 2 1 2 Sdl 0.04475617
#> 3 1 3 VSm 0.17891749
#> 4 1 4 Sm 0.19538752
#> 5 1 5 VLa 0.08122417
#> 6 1 6 Flo 0.11465228
#> 7 1 7 Dorm 0.05043913
Notice that the ss_prop column shows values that are a bit different from the ahistorical case, suggesting the influence of individual history. To see the impact of history on the stable stage distribution, let’s plot the ahistorical and historically-corrected stable stage distributions together.
ss_put_together <- cbind.data.frame(ehrlen2mss$ss_prop, ehrlen3mss$ahist$ss_prop)
names(ss_put_together) <- c("ahist", "hist")
rownames(ss_put_together) <- ehrlen2mss$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage", ylim = c(0, 0.35),
col = c("black", "red"), bty = "n")
legend("topright", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 2. Ahistorical vs. historically-corrected stable stage distribution
Accounting for individual history increased the prevalence of dormant seeds and small adults, but decreased the prevalence of very small and dormant adults. For example, dormant seeds are now expected to compose 33% of the population (29% in the pure ahistorical case), and very small and small adults together compose 37% of the population instead of 41%.
Let’s take a look at the reproductive values now, in similar order to the stable stage distribution case. Initially, we will create both sets of reproductive value objects.
The structure of these objects is the same as before, and, in the case of the ahistorical MPM, the reproductive values are also the same as those produced by popbio’s reproductive.value() function. Let’s compare the ahistorical vs. historically-corrected reproductive values of the life history stages. We will also standardize the reproductive values to make them comparable.
rv_put_together <- cbind.data.frame(ehrlen2mrv$rep_value, ehrlen3mrv$ahist$rep_value)
names(rv_put_together) <- c("ahist", "hist")
rv_put_together$ahist <- rv_put_together$ahist / max(rv_put_together$ahist)
rv_put_together$hist <- rv_put_together$hist / max(rv_put_together$hist)
rownames(rv_put_together) <- ehrlen2mrv$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topleft", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 3. Ahistorical vs. historically-corrected reproductive values
In both cases, flowering adults have the greatest reproductive value, while dormant seeds have the least. However, the historical MPM suggests a greater contribution of non-flowering but sprouting adult stages, and a lower contribution of vegetative dormancy.
Next we will look at the reproductive values of paired stages. Note that package popbio and R’s eigen() function may fail to estimate reproductive values in the historical case if analysis is attempted, because these approaches were not made to handle extremely large, sparse matrices. The final column (rep_value) of the $hist element is the reproductive value of the stage pair. We will plot these reproductive values.
Figure 4. Historical reproductive values
Examining these values (and comparing them against the list of paired stages included in $hist) reveals that the largest reproductive value is associated with flowering adults that were previously flowering, while the next largest reproductive value is associated with small adults that were previously small, and very small adults that were previously very small. Interestingly, historical transitions to very large adult have lower reproductive value here than we might expect given the output of the pure ahistorical case, where they were predicted to have the second highest reproductive value.
Now we can take a look at sensitivities of \(\lambda\) to the matrix elements. Here we conduct a sensitivity analysis on the ahistorical mean matrix.
ehrlen2sens <- sensitivity3(ehrlen2mean)
print(ehrlen2sens$sensmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.0182 0.00284 0.0142 0.0116 0.00479 0.00705 0.00347
#> [2,] 0.2061 0.03225 0.1611 0.1312 0.05434 0.07994 0.03936
#> [3,] 0.2565 0.04016 0.2006 0.1633 0.06766 0.09953 0.04900
#> [4,] 0.4110 0.06432 0.3213 0.2616 0.10838 0.15943 0.07849
#> [5,] 0.5639 0.08826 0.4408 0.3589 0.14871 0.21877 0.10770
#> [6,] 0.7221 0.11302 0.5645 0.4596 0.19043 0.28014 0.13791
#> [7,] 0.3067 0.04801 0.2398 0.1952 0.08089 0.11899 0.05858
writeLines("\nThe highest sensitivity value is: ")
#>
#> The highest sensitivity value is:
max(ehrlen2sens$sensmats[[1]])
#> [1] 0.7220817The highest sensitivity value is associated with a logical 0 - dormant seeds cannot transition to flowering adults, but the highest sensitivity value is associated with that element. This is an unfortunate issue in sensitivity analysis, and requires great care to prevent sloppy inference. Within the biologically plausible elements, the highest sensitivity appears to be associated with the transition from very small adult to flowering adult, with the second largest sensitivity associated with the transition from small adult to flowering adult.
We will now look at the sensitivity of \(\lambda\) to elements in the historical mean MPM. Because the matrices are full of 0s, we will look for the highest sensitivity associated with a non-zero matrix element. We will also not output the sensitivity matrices here, as they are quite large. Type ehrlen3sens at the prompt to see all sensitivity matrices in detail, and ehrlen3sens$h_sensmats to focus in just on historical sensitivities.
ehrlen3sens <- sensitivity3(ehrlen3mean)
writeLines("\nThe highest sensitivity value: ")
#>
#> The highest sensitivity value:
max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)])
#> [1] 0.2756227
writeLines("\n This value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen3sens$h_sensmats[[1]] == max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)]))
#> [1] 313Although we did not print it to the screen, the actual output is quite long because the matrices used are big. The first element produced is $h_sensmats, which is a list composed of sensitivity matrices of the historical matrices, in order (we have only one in our mean matrix object). These matrices are the same dimensions as the historical matrices used as input, and so can be quite huge. This is followed by $ah_sensmats, which is a list composed of historically-corrected sensitivity matrices of corresponding ahistorical matrix elements (calculated using the historically-corrected stable stage distribution and reproductive value vector produced in ehrlen3mss and ehrlen3mrv, respectively). So, these are different than the sensitivities estimated from the ahistorical matrices themselves, but have the same dimensions. Finally, $h_stages and $ah_stages give the order of paired stages and life history stages used in the historical and historically-corrected sensitivity matrices, respectively.
The maximum sensitivity value in the historical matrix appears to be associated with column 11 and row 13. This transition is from small adult in times t-1 and t to flowering adult in t+1. So, once again we see the importance of these two stages, but with greater resolution than was possible in the ahistorical case.
An alternative, or complementary, perturbation analysis to sensitivity analysis is elasticity analysis. Elasticities are easier to interpret because 0 elements produce 0 elasticity values, thus eliminating biologically impossible transitions from consideration, and because they are scaled to sum to 1.0. This scaling eliminates the units of the transition, making elasticities of survival transitions comparable to those of fecundity. However, they are also interpreted differently, because while sensitivity analysis shows the impact of a tiny but absolute change to a matrix element on \(\lambda\), elasticity analysis shows the impact of a tiny but proportional change to a matrix element on \(\lambda\). It is therefore not unusual for sensitivity and elasticity analysis to yield different inferences.
Here, we look at the elasticity of \(\lambda\) to matrix elements in the ahistorical mean matrix.
ehrlen2elas <- elasticity3(ehrlen2mean)
print(ehrlen2elas$elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.00655 0.00000 0.000000 0.0000 0.00000 0.0116 0.00000
#> [2,] 0.01162 0.00000 0.000000 0.0000 0.00000 0.0206 0.00000
#> [3,] 0.00000 0.02784 0.151678 0.0164 0.00052 0.0000 0.00415
#> [4,] 0.00000 0.00127 0.041102 0.1586 0.02210 0.0167 0.02184
#> [5,] 0.00000 0.00000 0.000604 0.0290 0.04851 0.0532 0.01744
#> [6,] 0.00000 0.00000 0.000000 0.0353 0.06862 0.1673 0.00887
#> [7,] 0.00000 0.00315 0.007180 0.0223 0.00896 0.0107 0.00628
writeLines("\nThe maximum elasticity value: ")
#>
#> The maximum elasticity value:
max(ehrlen2elas$elasmats[[1]])
#> [1] 0.167339Elasticity analysis reveals strong differences from sensitivity analysis here. In particular, we find that \(\lambda\) is most strongly elastic in response to changes in stasis transitions in flowering adults. We can sum the columns of the elasticity matrix to see which stages \(\lambda\) is most and least elastic in response to, as below.
print(colSums(ehrlen2elas$elasmats[[1]]), digits = 3)
#> [1] 0.0182 0.0323 0.2006 0.2616 0.1487 0.2801 0.0586Here we see that \(\lambda\) is most strongly elastic in response to changes in transitions associated with flowering adults, with transitions involving small adults coming in second. Dormant seeds and seedlings have the smallest impact on \(\lambda\), and the impacts of fecundity appear quite small.
Now to elasticity analysis of the historical MPMs. Once again, we will not output the matrices. Type ehrlen3elas at the prompt to see these matrices.
ehrlen3elas <- elasticity3(ehrlen3mean)
writeLines("\nThe highest elasticity value: ")
#>
#> The highest elasticity value:
max(ehrlen3elas$h_elasmats[[1]])
#> [1] 0.1429071
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen3elas$h_elasmats[[1]] == max(ehrlen3elas$h_elasmats[[1]]))
#> [1] 156The highest elasticity appears to be associated with row and column 6. This corresponds to the stasis transition of very small adults (very small in t-1 to very small in t to very small in t+1). Notice that the matrix is full of 0s, and that these correspond to the 0s in the original matrix.
Elasticities are additive, making the calculation of historically-corrected elasticity matrix easy. These are stored in the $ah_elasmats element of elasticity3() output originating from a historical MPM.
print(ehrlen3elas$ah_elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0 0 0.00000 0.0000 0.00000 0.0000 0.00000
#> [2,] 0 0 0.00000 0.0000 0.00000 0.0000 0.00000
#> [3,] 0 0 0.16285 0.0369 0.00013 0.0000 0.00261
#> [4,] 0 0 0.03816 0.1778 0.03268 0.0347 0.02064
#> [5,] 0 0 0.00000 0.0342 0.05415 0.0653 0.01057
#> [6,] 0 0 0.00000 0.0374 0.06812 0.1758 0.00532
#> [7,] 0 0 0.00152 0.0177 0.00906 0.0108 0.00349The historical matrices we used as input are reduced, but the historically-corrected output is in the original dimensions corresponding to the number of stages listed in the stageframe. This makes it relatively easy to compare against the actual ahistorical elasticity matrix, for example by subtracting one matrix from the other to see the differences.
print((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$elasmats[[1]]), digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] -0.00655 0.00000 0.000000 0.00000 0.000000 -0.01162 0.00000
#> [2,] -0.01162 0.00000 0.000000 0.00000 0.000000 -0.02063 0.00000
#> [3,] 0.00000 -0.02784 0.011174 0.02056 -0.000390 0.00000 -0.00154
#> [4,] 0.00000 -0.00127 -0.002939 0.01928 0.010574 0.01801 -0.00120
#> [5,] 0.00000 0.00000 -0.000604 0.00518 0.005642 0.01208 -0.00687
#> [6,] 0.00000 0.00000 0.000000 0.00208 -0.000497 0.00849 -0.00356
#> [7,] 0.00000 -0.00315 -0.005658 -0.00464 0.000100 0.00018 -0.00279Accounting for history shows \(\lambda\) to be less elastic in response to early-life transitions, transitions from dormancy, and fecundity than suggested by ahistorical matrices, but more elastic in response to changes in survival-transitions in small, very large, and flowering adults. We can also check to see where the maximum is located.
max(ehrlen3elas$ah_elasmats[[1]])
#> [1] 0.1778468
which(ehrlen3elas$ah_elasmats[[1]] == max(ehrlen3elas$ah_elasmats[[1]]))
#> [1] 25In the pure ahistorical analysis, we found that \(\lambda\) was most elastic in response to transitions from flowering adult to flowering adult. However, the historically-corrected analysis suggests that \(\lambda\) is most elastic in response to transitions from small adult to small adult, followed by stasis in the flowering adult stage. Some of these differences may stem from intra-class heterogeneity that is better captued in the historical MPM. For example, seedlings become small vegetative are less likely to move to the next larger stage than are plants that have remained as small vegetative for several years. The historical MPM would seperate these two instances, but the ahistorical MPM would treat all such transitions as originating only from the small vegetative stage. Once again, we see the impact of individual history.
Finally, we will look at a barplot of the elasticities of life history stages from ahistorical vs. historically-corrected analyses.
elas_put_together <- cbind.data.frame(colSums(ehrlen2elas$elasmats[[1]]), colSums(ehrlen3elas$ah_elasmats[[1]]))
names(elas_put_together) <- c("ahist", "hist")
rownames(elas_put_together) <- ehrlen2elas$stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity of lambda", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topleft", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 5. Ahistorical vs. historically-corrected elasticity of lambda to stages
The overall importance is similar across analyses in relative terms. However, ahistorical analysis shows that \(\lambda\) is most elastic in response to survival of flowering adults, while historically-corrected analysis suggests that it is most elastic in response to survival of small vegetative adults.
Further analytical tools are being planned for lefko3, but packages that handle projection matrices can typically handle the individual matrices produced and saved in lefkoMat objects in this package. Differences, obscure results, and errors sometimes arise when packages are not made to handle large and/or sparse matrices - historical matrices are both, and so care must be taken with their analysis.
Analysis 2: Function-based MPM estimation
In this analysis, we will build function-based MPMs using the Lathyrus dataset. To spice things up, we will use a slightly different approach to size classification, using the natural log of size instead of the normal size shown in the dataset. This has to do with the fact that volume is used as the size metric here, and so should have an allometric relationship to some vital rates (note that all size metrics have allometric relationships, but this is clearer when size is based on something more strongly related to mass, as is volume). We will also create both reproductive or non-reproductive stages of the same size classes.
Step 1. Life history model development
First, we will clear memory and reload the data, and then we will create a stageframe for the dataset. For this purpose, let’s look back at a summary of the main dataset, focusing at the distribution of log sizes.
rm(list=ls(all=TRUE))
data(lathyrus)
summary(c(lathyrus$Volume88, lathyrus$Volume89, lathyrus$Volume90, lathyrus$Volume91))
#> Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
#> 1.8 14.7 123.0 484.2 732.5 7032.0 1248
summary(c(lathyrus$lnVol88, lathyrus$lnVol89, lathyrus$lnVol90, lathyrus$lnVol91))
#> Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
#> 0.600 2.700 4.800 4.777 6.600 8.900 1248It is important to note the size minima and maxima, because we have been using 0 as the size of vegetatively dormant individuals. The lowest uncorrected size is 1.8, with a maximum above 7000. Since the minimum is above 1, all log sizes should be positive, and no given size value should result in an NA value when its logarithm is taken. Looking at the log sizes, we see all positive values, and the same number of NAs as in the original size distribution. This suggests that we are still able to use the value 0 as an indicator of vegetative dormancy, and that vegetative dormancy is currently included in the many NAs that occur in size variables in this dataset.
It can also help to take a look at plots of these distributions. First, we will look at the raw volume data.
plot(density(c(lathyrus$Volume88, lathyrus$Volume89, lathyrus$Volume90, lathyrus$Volume91),
na.rm = TRUE), main = "", xlab = "Volume", bty = "n")
Figure 6. Density plot of aboveground plant volume
Next we will look at the log volume data.
plot(density(c(lathyrus$lnVol88, lathyrus$lnVol89, lathyrus$lnVol90, lathyrus$lnVol91),
na.rm = TRUE), main = "", xlab = "Log volume", bty = "n")
Figure 7. Density plot of log aboveground plant volume
We see two very differently shaped distributions. The log volume distribution looks ‘better’ than the volume distribution, in the sense that it is closer to some semblance of a Gaussian distribution. This is helpful since our size metrics are decimals, and so cannot be treated as integers (in the latter case, we could try modeling them as either Poisson- or negative binomial-distributed). We will work with log volume in this example, and treat it as Gaussian-distributed.
We need to develop a stageframe that incorporates all log volumes that occur in the dataset, because our approach will include developing estimates of vital rates given inputs including all possible sizes, reproductive status, and so forth. We will build this by creating vectors of the values describing each stage, always in the same order.
sizevector <- c(0, 4.6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9)
stagevector <- c("Sd", "Sdl", "Dorm", "Sz1nr", "Sz2nr", "Sz3nr", "Sz4nr", "Sz5nr",
"Sz6nr", "Sz7nr", "Sz8nr", "Sz9nr", "Sz1r", "Sz2r", "Sz3r", "Sz4r",
"Sz5r", "Sz6r", "Sz7r", "Sz8r", "Sz9r")
repvector <- c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1)
obsvector <- c(0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
matvector <- c(0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
immvector <- c(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
indataset <- c(0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
binvec <- c(0, 4.6, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5)
lathframeln <- sf_create(sizes = sizevector, stagenames = stagevector, repstatus = repvector,
obsstatus = obsvector, matstatus = matvector, immstatus = immvector,
indataset = indataset, binhalfwidth = binvec, propstatus = propvector)
To be useful to others who might try to understand your work, or to yourself in the future, it helps to add text descriptions of the stages. Here, we add some stage descriptions to this stageframe in the comments field.
lathframeln$comments <- c("Dormant seed", "Seedling", "Dormant", "Size 1 Non-reprod", "Size 2 Non-reprod",
"Size 3 Non-reprod", "Size 4 Non-reprod", "Size 5 Non-reprod", "Size 6 Non-reprod",
"Size 7 Non-reprod", "Size 8 Non-reprod", "Size 9 Non-reprod", "Size 1 Reprod",
"Size 2 Reprod", "Size 3 Reprod", "Size 4 Reprod", "Size 5 Reprod", "Size 6 Reprod",
"Size 7 Reprod", "Size 8 Reprod", "Size 9 Reprod")Step 2a. Data organization
Once the stageframe is created, we can reorganize the dataset into vertical format. Here, ‘vertical’ format is a way of organizing demographic data in which each row corresponds to the state of a single individual in two (if ahistorical) or three (if historical) consecutive times. To handle this, we use the verticalize3() function, which creates historically-formatted vertical datasets, as below. We also need to get rid of NAs in size and fecundity variables for modelsearch to work properly when we actually build models of vital rates, so we will now use the NAas0 = TRUE option. Finally, note that we are using a fecundity variable, Intactseed88, to denote both fecundity and reproductive status. Although this dataset includes a term that codes for whether a plant flowered (FCODE88, which was used in the previous example with raw matrices), it includes aborted flowering attempts in the results, leading to 0s in fecundity for plants that attempted to reproduce. Using FCODE88 would mean incorporating these 0s into the fecundity variable, treating flowering rather than fruiting as reproduction. That may be warranted, because flowering might have costs, but it could also have unintended consequences. For example, if we choose to model fecundity under a Gaussian distribution, then negative fecundity estimates may occur. If we choose a Poisson or negative binomial distribution for fecundity, then all 0s will be dropped from modeling and we will get a warning during the modelsearch() routine. So, the choice of which variable to use for reproductive status should be given a great deal of consideration. Here, we do not include 0s in fecundity, choosing the more appropriate reproductive status term, Intactseed88.
lathvertln <- verticalize3(lathyrus, noyears = 4, firstyear = 1988, patchidcol = "SUBPLOT",
individcol = "GENET", blocksize = 9, juvcol = "Seedling1988",
size1col = "lnVol88", repstr1col = "Intactseed88",
fec1col = "Intactseed88", dead1col = "Dead1988",
nonobs1col = "Dormant1988", stageassign = lathframeln,
stagesize = "sizea", censorcol = "Missing1988",
censorkeep = NA, NAas0 = TRUE, censor = TRUE)Something to note here is that fecundity is typically but not always an integer. We can see this below, where we show each non-integer in the dataset. There are a total of 6 out of 2,527 actual transitions in the dataset. These 6 cases are not errors - they are due to the fact that fecundity was estimated in two ways in this dataset, one way leading to integers, and one leading to decimals (Ehrlén 2000).
lathvertln$feca2[which(lathvertln$feca2 != round(lathvertln$feca2))]
#> [1] 12.500000 3.818182 26.600000 23.333333 5.333333 24.500000
length(lathvertln$feca2)
#> [1] 2527We can either treat fecundity as a continuous variable, or round the values and treat them as count variables. In this dataset, fecundity is mostly counts of intact seeds, and only differs in six cases where the seed output was estimated based on other models. So, we will round fecundity so that we can assume count variables in the analysis, as below.
lathvertln$feca2 <- round(lathvertln$feca2)
lathvertln$feca1 <- round(lathvertln$feca1)
lathvertln$feca3 <- round(lathvertln$feca3)All set! Now to move on to supplemental descriptive information.
Step 2b. Provide supplemental information for matrix estimation
Next we will create a reproductive matrix, which describes which stages are reproductive, which stages they produce, and at what levels they reproduce. This matrix is mostly composed of 0s, but fecundity is noted as non-zero entries equal to a scalar multiplier to the full fecundity estimated by lefko3. This matrix has rows and columns equal to the number of stages described in the stageframe for this dataset, and the rows and columns refer to these stages in the same order as in the stageframe. In many ways, it looks like a nearly empty population matrix, but notes the per-individual mean multipliers on fecundity for each stage that actually reproduces. Here, we first create a square zero matrix with dimensions equal to the number of rows in lathframeln. Then we modify elements corresponding to fecundity by the expected mean seed dormancy probability (row 1), and by the germination rate (row 2).
Next we need to provide some given transitions. We will use the same overwrite tables as in the last example (objects lathover2 and lathover3), because these relate only to stages that are the same in both analyses. Here they are again.
lathover2 <- overwrite(stage3 = c("Sd", "Sdl"), stage2 = c("Sd", "Sd"),
givenrate = c(0.345, 0.054))
lathover3 <- overwrite(stage3 = c("Sd", "Sd", "Sdl"), stage2 = c("Sd", "Sd", "Sd"),
stage1 = c("Sd", "rep", "rep"), givenrate = c(0.345, 0.345, 0.054))Next, we move to a new step: modeling vital rates.
Step 3. Tests of history, and vital rate modeling
Now we will run the modelsearch function with the new vertical dataset, first for the ahistorical model set and then for the historical model set. This function will develop our best-fit vital rate models for us, and also allow us to test whether individual history is a significant influence on vital rates. This function looks simple, but it automates several crucial and complex tasks in MPM analysis. Specifically, it automates 1) the building of global models for each vital rate requested, 2) the exhaustive construction of all reduced models, and 3) the selection of the best-fit models.
Let’s look at some of the options that we will utilize for this purpose (please note that this list includes only some of the options actually offered by the function - further options are shown in the documentation for modelsearch(), and further theoretical details are shown in the Basic theory and concepts vignette).
historical: Setting this to TRUE or FALSE tells R whether to include state in time t-1 in the global models.
approach: This option tells R whether to use generalized linear models (glm), in which all factors are treated as fixed, or mixed effects models (lme4), in which factors are treated as either fixed or random (most notably, time, patch, and individual identity can be treated as random). We encourage users to use the latter option, as it accounts for pseudoreplication, but the former approach is more common.
suite: This tells R which factors to include in the global model. Possible values include size, which includes size only; rep, which includes reproductive status only; main, which includes both size and reproductive status as main effects only; full, which includes both size and reproductive status and all two-way interactions; and const, which includes only an intercept. These factors are in addition to individual identity, patch, and time.
vitalrates: This option tells R which vital rates to model. The default is to model survival, size, and fecundity, but users can also model observation status and reproduction status.
juvestimate: This optional term tells R the name of the juvenile stage transitioning to maturity, in cases where the dataset includes data on juveniles.
juvsize: This optional term denotes whether size should be used as an independent term in models involving transition from the juvenile stage.
bestfit: This describes whether the best-fit model should be chosen as the model with the lowest AICc (AICc) or as the most parsimonious model (AICc&k), where the latter is the model that has the fewest estimated parameters and is within 2 AICc units of the model with the lowest AICc. We encourage users to choose the latter. This is the default setting, and is more in-line with model selection protocols preferred by information theorists (Burnham and Anderson 2002)..
sizedist: This designates the distribution used for size. The options include gaussian, poisson, and negbin, the last of which refers to the negative binomial distribution.
fecdist: This designates the distribution used for fecundity. The options include gaussian, poisson, and negbin, the last of which refers to the negative binomial distribution.
fectime: This designates whether fecundity is estimated within time t (the default) or time t+1. Plant ecologists are likely to choose the former, since fecundity is typically estimated via a proxy such as flowers, fruit, or seed produced. Wildlife ecologists might choose the latter, since fecundity may be best estimated as a count of actual juveniles within nests, burrows, or other family structures.
indiv: This should include the name of the variable corresponding to individual identity in the vertical dataset.
patch: This should include the name of the variable corresponding to patch identity in the vertical dataset.
year: This should include the name of the variable corresponding to time t in the vertical dataset.
year.as.random: This tells R whether to treat year as a random factor, and is only used when approach = "lme4".
patch.as.random: This tells R whether to treat patch identity as a random factor, and is only used when approach = "lme4".
show.model.tables: This tells R to include the full dredge model tables showing all models developed and their associated AICc values.
quiet: This tells R to silence guidepost statements and most warning messages. Only warnings incurred in model testing will be visible, including warnings from the testing of global models and those from the testing of reduced models produced during model dredging.
In addition, there are other options that provide flexibility in handling datasets with different designations for key variables, and allowing manual designation of stages.
This function is useful not just because it develops the models that we can use to estimate function-based MPMs, but because it provides a test of whether individual history affects demography. Setting historical = TRUE fits size and/or reproductive status in time t-1 into global models. If any of the resulting best-fit vital rate models include one of these terms, then we see a significant effect of individual history on demography, and we have a reason to proceed with historical MPM analysis. Here, we will build both ahistorical and historical models for comparison.
First we will create the ahistorical model set.
lathmodelsln2 <- modelsearch(lathvertln, historical = FALSE, approach = "lme4", suite = "full",
vitalrates = c("surv", "obs", "size", "repst", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "poisson",
indiv = "individ", patch = "patchid", year = "year2",
year.as.random = TRUE, patch.as.random = TRUE, quiet = TRUE)
#> boundary (singular) fit: see ?isSingular
#> fixed-effect model matrix is rank deficient so dropping 2 columns / coefficients
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
The output can be rather verbose, and so we have limited it with the quiet = TRUE option. The function was developed to provide text marker posts of what the function is doing and what it has accomplished, as well as to show all warnings from all workhorse functions used. Because we used the lme4 approach here, this includes warnings originating from estimating mixed linear models with package lme4 (Bates et al. 2015). It also shows warnings originating from the dredge() function of package MuMIn (Bartoń 2014), which is the core function used in model building and AICc estimation. We encourage users to get familiar with interpreting these warnings and assessing the degree to which they impact their own analyses.
In developing these linear models, the modelsearch() function set up a series of nested datasets for use in estimating vital rates as conditional rates and probabilities. The exact subsets created depend on which parameters are called for. It is important to consider which are required, and particular attention needs to be paid if there are size classes with sizes of 0. In these situations, it is best to consider a size of 0 as unobservable, and so to introduce observation status as a vital rate. This sets up datasets for size estimation that do not include 0 in the response term, because all 0 responses are already absorbed by observation status.
Let’s take a peek at the summary for the model selection output. This condenses the output to the necessities.
summary(lathmodelsln2)
#> This LefkoMod object includes 8 linear models.
#> Best-fit model criterion used: AICc&k
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea2 + (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 931.9880 960.5725 -460.9940 921.9880 2241
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.55492
#> patchid (Intercept) 0.28264
#> year2 (Intercept) 0.05861
#> Number of obs: 2246, groups: individ, 257; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 1.8559 0.2455
#>
#> ────────────────────────────────────────
#>
#> Observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: obs.data
#> AIC BIC logLik deviance df.resid
#> 1342.9524 1365.5910 -667.4762 1334.9524 2117
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 6.46e-05
#> patchid (Intercept) 3.61e-01
#> year2 (Intercept) 0.00e+00
#> Number of obs: 2121, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.24
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ sizea2 + (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: size.data
#> REML criterion at convergence: 6042.594
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.6538
#> patchid (Intercept) 0.1917
#> year2 (Intercept) 0.3213
#> Residual 1.0734
#> Number of obs: 1916, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 2.4325 0.4843
#>
#> ────────────────────────────────────────
#>
#> Reproductive status model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: repstatus3 ~ sizea2 + (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: repst.data
#> AIC BIC logLik deviance df.resid
#> 1187.2858 1215.0757 -588.6429 1177.2858 1911
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.3210
#> patchid (Intercept) 0.2617
#> year2 (Intercept) 0.9822
#> Number of obs: 1916, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> -7.8614 0.9814
#>
#> ────────────────────────────────────────
#>
#> Fecundity model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: poisson ( log )
#> Formula: feca2 ~ sizea2 + (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: fec.data
#> AIC BIC logLik deviance df.resid
#> 2053.538 2071.455 -1021.769 2043.538 261
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.5448
#> patchid (Intercept) 0.1236
#> year2 (Intercept) 0.1502
#> Number of obs: 266, groups: individ, 101; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> -2.1824 0.5999
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Juvenile survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvsurv.data
#> AIC BIC logLik deviance df.resid
#> 323.9167 338.4701 -157.9583 315.9167 277
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.0000
#> patchid (Intercept) 0.3623
#> year2 (Intercept) 0.0000
#> Number of obs: 281, groups: individ, 187; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 1.07
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvobs.data
#> AIC BIC logLik deviance df.resid
#> 93.4925 106.8809 -42.7462 85.4925 206
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 16.009
#> patchid (Intercept) 0.000
#> year2 (Intercept) 1.221
#> Number of obs: 210, groups: individ, 154; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 10.39
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvsize.data
#> REML criterion at convergence: 243.3232
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.08956
#> patchid (Intercept) 0.00000
#> year2 (Intercept) 0.09066
#> Residual 0.43789
#> Number of obs: 193, groups: individ, 144; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.29
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile reproduction model:
#> [1] 0
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Number of models in survival table:5
#>
#> Number of models in observation table:5
#>
#> Number of models in size table:5
#>
#> Number of models in reproduction status table:5
#>
#> Number of models in fecundity table:5
#>
#> Number of models in juvenile survival table:1
#>
#> Number of models in juvenile observation table:1
#>
#> Number of models in juvenile size table:1
#>
#> Number of models in juvenile reproduction table: 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> General model parameter names (column 1), and specific names used in these models (column 2):
#> mainparams modelparams
#> 1 year2 year2
#> 2 individ individ
#> 3 patch patchid
#> 4 surv3 alive3
#> 5 obs3 obsstatus3
#> 6 size3 sizea3
#> 7 repst3 repstatus3
#> 8 fec3 feca3
#> 9 fec2 feca2
#> 10 size2 sizea2
#> 11 size1 <NA>
#> 12 repst2 repstatus2
#> 13 repst1 <NA>
#> 14 age <NA>
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLThe best-fit models vary a bit in complexity. For example, survival is influenced by size in the current year, as well as by patch, individual identity, and year, while observation status is not influenced by size. We can see these models explicitly, as well as the model tables developed, by calling them directly from the lefkoMod object. Here is the best-fit model for survival.
lathmodelsln2$survival_model
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea2 + (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 931.9880 960.5725 -460.9940 921.9880 2241
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.55492
#> patchid (Intercept) 0.28264
#> year2 (Intercept) 0.05861
#> Number of obs: 2246, groups: individ, 257; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 1.8559 0.2455
Now let’s estimate the historical model set. This search will take more time, because the inclusion of status in time t-1 (and all two-way interactions when suite = "full") increases the number of models that the dredge function will build and compare. The inclusion of these terms may also create more warnings as the function proceeds. Please also note that we include quiet = TRUE here, which prevents the model selection protocol from issuing warnings and most diagnostic messages. We do this only because the model selection will generate a particularly verbose list of warnings in this case. Normally it is best to stick with the default, quiet = FALSE.
lathmodelsln3 <- modelsearch(lathvertln, historical = TRUE, approach = "lme4", suite = "full",
vitalrates = c("surv", "obs", "size", "repst", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "poisson",
indiv = "individ", patch = "patchid", year = "year2",
year.as.random = TRUE, patch.as.random = TRUE,
show.model.tables = TRUE, quiet = TRUE)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00559824 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: large eigenvalue ratio
#> - Rescale variables?
#> boundary (singular) fit: see ?isSingular
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.165108 (tol =
#> 0.002, component 1)
#> fixed-effect model matrix is rank deficient so dropping 4 columns / coefficients
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.559612 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: very large eigenvalue
#> - Rescale variables?;Model is nearly unidentifiable: large eigenvalue ratio
#> - Rescale variables?
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
summary(lathmodelsln3)
#> This LefkoMod object includes 8 linear models.
#> Best-fit model criterion used: AICc&k
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea1 + sizea2 + (1 | individ) + (1 | year2) + (1 | patchid) + sizea1:sizea2
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 897.7195 937.7379 -441.8598 883.7195 2239
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.2799
#> patchid (Intercept) 0.2720
#> year2 (Intercept) 0.4861
#> Number of obs: 2246, groups: individ, 257; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) sizea1 sizea2 sizea1:sizea2
#> 0.9250 0.4453 0.3187 -0.0453
#>
#> ────────────────────────────────────────
#>
#> Observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: obs.data
#> AIC BIC logLik deviance df.resid
#> 1342.9524 1365.5910 -667.4762 1334.9524 2117
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 6.46e-05
#> patchid (Intercept) 3.61e-01
#> year2 (Intercept) 0.00e+00
#> Number of obs: 2121, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.24
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ repstatus1 + repstatus2 + sizea1 + sizea2 + (1 | individ) +
#> (1 | year2) + (1 | patchid) + repstatus1:sizea1 + repstatus2:sizea2 + sizea1:sizea2
#> Data: size.data
#> REML criterion at convergence: 5298.341
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.1861
#> patchid (Intercept) 0.1340
#> year2 (Intercept) 0.5063
#> Residual 0.9356
#> Number of obs: 1916, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) repstatus1 repstatus2 sizea1 sizea2 repstatus1:sizea1 repstatus2:sizea2
#> 0.35241 -3.35781 -1.84618 0.65491 0.81505 0.53176 0.29185
#> sizea1:sizea2
#> -0.09023
#>
#> ────────────────────────────────────────
#>
#> Reproductive status model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: repstatus3 ~ repstatus2 + sizea1 + sizea2 + (1 | individ) + (1 | year2) + (1 | patchid) + sizea1:sizea2
#> Data: repst.data
#> AIC BIC logLik deviance df.resid
#> 1146.334 1190.798 -565.167 1130.334 1908
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.3053
#> patchid (Intercept) 0.2539
#> year2 (Intercept) 0.9475
#> Number of obs: 1916, groups: individ, 254; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) repstatus2 sizea1 sizea2 sizea1:sizea2
#> -12.0767 0.4135 1.1697 1.5343 -0.1618
#>
#> ────────────────────────────────────────
#>
#> Fecundity model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: poisson ( log )
#> Formula: feca2 ~ repstatus1 + sizea1 + sizea2 + (1 | individ) + (1 | year2) + (1 | patchid) + repstatus1:sizea1
#> Data: fec.data
#> AIC BIC logLik deviance df.resid
#> 2048.713 2077.381 -1016.357 2032.714 258
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.5400
#> patchid (Intercept) 0.1258
#> year2 (Intercept) 0.0000
#> Number of obs: 266, groups: individ, 101; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept) repstatus1 sizea1 sizea2 repstatus1:sizea1
#> -2.35725 1.80633 0.03454 0.60138 -0.23774
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Juvenile survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvsurv.data
#> AIC BIC logLik deviance df.resid
#> 323.9167 338.4701 -157.9583 315.9167 277
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.0000
#> patchid (Intercept) 0.3623
#> year2 (Intercept) 0.0000
#> Number of obs: 281, groups: individ, 187; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 1.07
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvobs.data
#> AIC BIC logLik deviance df.resid
#> 93.4925 106.8809 -42.7462 85.4925 206
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 16.009
#> patchid (Intercept) 0.000
#> year2 (Intercept) 1.221
#> Number of obs: 210, groups: individ, 154; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 10.39
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ (1 | individ) + (1 | year2) + (1 | patchid)
#> Data: juvsize.data
#> REML criterion at convergence: 243.3232
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.08956
#> patchid (Intercept) 0.00000
#> year2 (Intercept) 0.09066
#> Residual 0.43789
#> Number of obs: 193, groups: individ, 144; patchid, 6; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.29
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile reproduction model:
#> [1] 0
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Number of models in survival table:113
#>
#> Number of models in observation table:113
#>
#> Number of models in size table:113
#>
#> Number of models in reproduction status table:113
#>
#> Number of models in fecundity table:113
#>
#> Number of models in juvenile survival table:1
#>
#> Number of models in juvenile observation table:1
#>
#> Number of models in juvenile size table:1
#>
#> Number of models in juvenile reproduction table: 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> General model parameter names (column 1), and specific names used in these models (column 2):
#> mainparams modelparams
#> 1 year2 year2
#> 2 individ individ
#> 3 patch patchid
#> 4 surv3 alive3
#> 5 obs3 obsstatus3
#> 6 size3 sizea3
#> 7 repst3 repstatus3
#> 8 fec3 feca3
#> 9 fec2 feca2
#> 10 size2 sizea2
#> 11 size1 sizea1
#> 12 repst2 repstatus2
#> 13 repst1 repstatus1
#> 14 age <NA>
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLThis summary reveals some similarities, but also many differences. Historical models incorporate a further time back in the global model, so some of the best-fit models may have historical terms in them. Here, we see some of these historical terms in the models of survival, size, reproduction status, and fecundity. We also see that incorporating a time back ends up changing the model best-fit model structure such that reproductive status also becomes an important determinant of some vital rates, even where it was absent in the ahistorical case. This suggests that individual history is important to the demography of this population, and should not be ignored.
Now we are ready to estimate matrices.
Step 4. MPM estimation
For instructional purposes, we will make both ahistorical and historical MPMs. We will start by estimating the ahistorical set of matrices. Notice that we are matching the ahistorical matrix creation function, flefko2(), with the appropriate ahistorical input, including the ahistorical lefkoMod object lathmodelsln2. Model sets that include historical terms should not be used to create ahistorical matrices, since the coefficients in the best-fit models are estimated assuming a specific model structure. Historical vital rate models may yield biased results if used to construct ahistorical matrices.
lathmat2ln <- flefko2(stageframe = lathframeln, modelsuite = lathmodelsln2,
data = lathvertln, repmatrix = lathrepmln, overwrite = lathover2,
year.as.random = FALSE, patch.as.random = FALSE)
summary(lathmat2ln)
#>
#> This ahistorical lefkoMat object contains 18 matrices.
#>
#> Each matrix is a square matrix with 21 rows and columns, and a total of 441 elements.
#> A total of 6732 survival transitions were estimated, with 374 per matrix.
#> A total of 324 fecundity transitions were estimated, with 18 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLLet’s take a look at a portion of the first of these matrices.
print(lathmat2ln$A[[1]][,1:5], digits = 3)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.345 0.00e+00 0.00e+00 0.00e+00 0.00e+00
#> [2,] 0.054 5.99e-06 0.00e+00 0.00e+00 0.00e+00
#> [3,] 0.000 5.28e-05 6.55e-02 6.71e-02 6.84e-02
#> [4,] 0.000 9.91e-03 1.01e-01 4.77e-02 1.83e-02
#> [5,] 0.000 6.01e-01 2.62e-01 1.88e-01 1.10e-01
#> [6,] 0.000 1.98e-01 2.86e-01 3.13e-01 2.77e-01
#> [7,] 0.000 3.54e-04 1.31e-01 2.18e-01 2.94e-01
#> [8,] 0.000 3.45e-09 2.51e-02 6.36e-02 1.31e-01
#> [9,] 0.000 1.82e-16 2.02e-03 7.80e-03 2.44e-02
#> [10,] 0.000 5.22e-26 6.84e-05 4.02e-04 1.92e-03
#> [11,] 0.000 8.14e-38 9.71e-07 8.69e-06 6.31e-05
#> [12,] 0.000 6.90e-52 5.79e-09 7.90e-08 8.72e-07
#> [13,] 0.000 0.00e+00 5.98e-05 7.53e-05 7.69e-05
#> [14,] 0.000 0.00e+00 1.55e-04 2.98e-04 4.63e-04
#> [15,] 0.000 0.00e+00 1.69e-04 4.93e-04 1.17e-03
#> [16,] 0.000 0.00e+00 7.73e-05 3.43e-04 1.24e-03
#> [17,] 0.000 0.00e+00 1.48e-05 1.00e-04 5.51e-04
#> [18,] 0.000 0.00e+00 1.20e-06 1.23e-05 1.03e-04
#> [19,] 0.000 0.00e+00 4.05e-08 6.35e-07 8.07e-06
#> [20,] 0.000 0.00e+00 5.75e-10 1.37e-08 2.66e-07
#> [21,] 0.000 0.00e+00 3.43e-12 1.25e-10 3.67e-09The number of elements estimated is vastly greater now than in the raw ahMPMs estimated in Analysis 1, because each matrix element must be estimated based on the vital rate linear models supplied. This matrix is overwhelmingly composed of elements that are to be estimated (392 / 441 = 88.9% of elements), whereas in the raw matrix case, elements would only be estimated if individuals actually passed through that transition (82.5 / 441 = 18.7% of elements estimated).
Now we will estimate the historical matrices.
lathmat3ln <- flefko3(stageframe = lathframeln, modelsuite = lathmodelsln3,
data = lathvertln, repmatrix = lathrepmln, overwrite = lathover3,
year.as.random = FALSE, patch.as.random = FALSE)
summary(lathmat3ln)
#>
#> This historical lefkoMat object contains 18 matrices.
#>
#> Each matrix is a square matrix with 441 rows and columns, and a total of 194481 elements.
#> A total of 130500 survival transitions were estimated, with 7250 per matrix.
#> A total of 6804 fecundity transitions were estimated, with 378 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULL
Once again, the matrices are vastly bigger than their ahistorical counterparts. The ahistorical matrices here are 21 rows x 21 columns, so contain 441 elements each. The historical matrices are 212 x 212 (441 x 441), or 194,481 elements large. However, because of the prevalence of structural 0s in the latter, only 7,628 elements are actually estimated (7,628 / 194,481 = 3.9% of elements estimated). This is still many more elements than the average 82.5 elements (18.7% of elements estimated) that were estimated per raw historical matrix in Analysis 1, but represents a smaller proportion. Another point of interest here is that the number of A matrices estimated is 18, which is the same as the number of ahistorical matrices. In the raw matrix case, there would actually only be 12 historical matrices estimated, because of the lack of information for the time previous to the start of the dataset in each of the 6 patches. Here, linear modeling treating year as random allows us to make assumptions leading to estimated values for even those first transitions. In particular, by setting year.as.random = FALSE, the default value for the year coefficients in years without random coefficients estimated is 0, while it would be a randomly chosen coefficient from the distribution of random year coefficients if year.as.random = TRUE.
Because of the size of the matrices and the proliferation of structural 0s, we may wish to analyze these matrices after removing some unneeded rows and columns. Let’s re-do the above, but overwrite the models with smaller matrices using the reduce = TRUE option, which eliminates stages or stage-pairs with corresponding row and column vectors both equal to zero vectors.
lathmat3ln <- flefko3(stageframe = lathframeln, modelsuite = lathmodelsln3,
data = lathvertln, repmatrix = lathrepmln, overwrite = lathover3,
year.as.random = FALSE, patch.as.random = FALSE, reduce = TRUE)
summary(lathmat3ln)
#>
#> This historical lefkoMat object contains 18 matrices.
#>
#> Each matrix is a square matrix with 410 rows and columns, and a total of 168100 elements.
#> A total of 130500 survival transitions were estimated, with 7250 per matrix.
#> A total of 6804 fecundity transitions were estimated, with 378 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLThis exercise eliminated 31 rows and columns, so the matrices definitely got smaller although they are still quite large. The number of estimated elements, however, is the same as before.
Now we will estimate the mean matrices. Note that these include 6 mean matrices for the patches, followed by a grand mean matrix, yielding a total of 7 matrices per lefkoMat object.
lathmat2lnmean <- lmean(lathmat2ln)
lathmat3lnmean <- lmean(lathmat3ln)
summary(lathmat2lnmean)
#>
#> This ahistorical lefkoMat object contains 7 matrices.
#>
#> Each matrix is a square matrix with 21 rows and columns, and a total of 441 elements.
#> A total of 2618 survival transitions were estimated, with 374 per matrix.
#> A total of 126 fecundity transitions were estimated, with 18 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULL
summary(lathmat3lnmean)
#>
#> This historical lefkoMat object contains 7 matrices.
#>
#> Each matrix is a square matrix with 410 rows and columns, and a total of 168100 elements.
#> A total of 50750 survival transitions were estimated, with 7250 per matrix.
#> A total of 2646 fecundity transitions were estimated, with 378 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproductive status estimated with 254 individuals and 1916 individual transitions.
#> Fecundity estimated with 101 individuals and 266 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLNext we will check the survival of stages and stage pairs, making sure that they appear to be probabilities. We will focus on the grand mean U matrix in each case.
writeLines("Stage survival in ahistorical grand mean matrix")
#> Stage survival in ahistorical grand mean matrix
summary(colSums(lathmat2lnmean$U[[7]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.3990 0.9024 0.9396 0.9002 0.9699 0.9763
writeLines("\nStage survival in historical grand mean matrix")
#>
#> Stage survival in historical grand mean matrix
summary(colSums(lathmat3lnmean$U[[7]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.0000 0.9153 0.9634 0.8552 0.9765 0.9913Quick scans of these numbers show no real problems. Note that the spread of values is greater in the historical case - this is expected, since the ahistorical transition values are partially determined by the historical transition values. We encourage finer-scale examination of each matrix for further error-checking.
Step 5. MPM analysis
Now let’s estimate the deterministic population growth rate \(\lambda\) in for historical and ahistorical MPM means and plot them. First, the ahistorical case. Remember that there are 6 patches, and the 7th \(\lambda\) in each case is \(\lambda\) for the population-level grand mean. So, we will just plot the 6 patch-level means.
lambda2lnm <- lambda3(lathmat2lnmean)
lambda3lnm <- lambda3(lathmat3lnmean)
plot(lambda ~ patch, data = lambda2lnm[1:6,], ylim = c(0.90, 1.00), type = "l", lty= 1, lwd = 2, bty = "n")
lines(lambda ~ patch, data = lambda3lnm[1:6,], lty = 2, lwd= 2, col = "red")
legend("bottomleft", c("ahistorical", "historical"), lty = c(1, 2), col = c("black", "red"), lwd = 2, bty = "n")
Figure 8. Ahistorical vs. historical lambda
These lambda estimates are all less than 1.0, suggesting a declining population. Qualitatively, they agree with the \(\lambda\) estimates from raw matrix analysis (Analysis 1). However, it is also clear that the hMPM estimates are less variable. This may be because of long-term trade-offs such as the cost of growth (Shefferson, Warren, and Pulliam 2014), although only further research can explain it properly.
Now let’s look at the stable stage distributions for both ahistorical and historical MPMs, using the population mean matrix. First, we will estimate the full distributions, and then compare the ahistorical stable stage distribution to the historically-corrected distribution. Note that our output for the ahistorical MPM match those produced by package popbio’s stable.stage() function.
ehrlen2ss <- stablestage3(lathmat2lnmean)
ehrlen3ss <- stablestage3(lathmat3lnmean)
ss_put_together <- cbind.data.frame(subset(ehrlen2ss, matrix == 7)$ss_prop,
subset(ehrlen3ss$ahist, matrix == 7)$ss_prop)
names(ss_put_together) <- c("ahist", "hist")
rownames(ss_put_together) <- subset(ehrlen2ss, matrix == 7)$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage", ylim = c(0, 0.35),
col = c("black", "red"), bty = "n")
legend("topright", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 9. Ahistorical vs. historically-corrected stable stage distribution
The output for the ahistorical MPM is a dataframe with matrix of origin, stage name, and stable stage proportion in each row. We have 7 sets of stable stage distributions because there are 7 matrices. Note that the last matrix, the 7th, is the grand mean matrix, and that the ss_prop column, which gives us the stable stage proportion of each stage, sums to 1.0 within each matrix. In the historical case, output includes 2 data frames, the first ($hist) of which is the full historical stage-pair distribution, and the second ($ahist) is the historically-corrected stable stage distribution.
The ahistorical and historically-corrected distributions differ quite a bit. For example, dormant seeds make up 31% of the population in the historically-corrected stable stage distribution, but only 12% in the ahistorical case. Seedlings make up only 4% of the historically-corrected stable stage distribution but 2% in the ahistorical case. Once again, we see an impact of individual history on an important characteristic of the dynamics of this population.
Looking at the full historical stable stage distribution, we see that the large proportion of the population corresponds to dormant seeds that continue as dormant seeds. The next largest proportions are of seeds produced by adults in reproductive size classes 6, 7, and 8, and these are followed by non-reproductive adults staying in size classes 4, 5, and 6.
barplot(subset(ehrlen3ss$hist, matrix == 7)$ss_prop, main = "Historical", ylab = "Proportion", xlab = "Stage-pair")
Figure 10. Historical stable stage distribution
Now let’s take a look at the reproductive values. We’ll go straight to the plots, as with the stable stage distribution.
ehrlen2rv <- repvalue3(lathmat2lnmean)
ehrlen3rv <- repvalue3(lathmat3lnmean)
rv_put_together <- cbind.data.frame(subset(ehrlen2rv, matrix == 7)$rep_value,
subset(ehrlen3rv$ahist, matrix == 7)$rep_value)
names(rv_put_together) <- c("ahist", "hist")
rv_put_together$ahist <- rv_put_together$ahist / max(rv_put_together$ahist)
rv_put_together$hist <- rv_put_together$hist / max(rv_put_together$hist)
rownames(rv_put_together) <- subset(ehrlen2rv, matrix == 7)$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topleft", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 11. Ahistorical vs. historically-corrected reproductive values
In the ahistorical case, the highest reproductive value appears to be associated with the largest flowering stage, Sz9r, while the lowest is associated with dormant seeds (as before, our rep_value estimates are similar to those estimated by popbio’s reproductive.value() function). These results are different from the historically-corrected case. Dormant seeds and seedlings appear to have essentially no reproductive value, and the smallest non-zero reproductive value is associated with Sz1r, the smallest flowering adult. The highest reproductive value is now Sz9nr, the largest non-flowering adult. We leave it to the reader to consider the importance of such differences.
We will next assess which matrix elements \(\lambda\) is most sensitive and elastic to changes in. First the sensitivity of the grand mean ahistorical MPM.
lathmat2msens <- sensitivity3(lathmat2lnmean)
writeLines("\nThe highest sensitivity value: ")
#>
#> The highest sensitivity value:
max(lathmat2msens$sensmats[[7]][which(lathmat2lnmean$A[[7]] > 0)])
#> [1] 0.7005325
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(lathmat2msens$sensmats[[7]] == max(lathmat2msens$sensmats[[7]][which(lathmat2lnmean$A[[7]] > 0)]))
#> [1] 147The highest sensitivity value appears to be associated with the transition from 4-sprouted non-flowering adult to the largest flowering adult. Inspecting the sensitivity matrix (type lathmat2msens$sensmats[[7]] to inspect the full matrix) also shows that transitions near that element in the matrix are also associated with rather high sensitivities.
Now we can compare the historical case, particularly focusing on the historically-corrected sensitivities.
lathmat3msens <- sensitivity3(lathmat3lnmean)
writeLines("\nThe highest sensitivity value: ")
#>
#> The highest sensitivity value:
max(lathmat3msens$ah_sensmats[[7]][which(lathmat2lnmean$A[[7]] > 0)])
#> [1] 8.284942
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(lathmat3msens$ah_sensmats[[7]] == max(lathmat3msens$ah_sensmats[[7]][which(lathmat2lnmean$A[[7]] > 0)]))
#> [1] 168Here, the highest elasticity value is associated with transition from 5-sprouted non-flowering adult (Sz5nr) to the largest flowering adult, making it similar but not exactly equal to the ahistorical case. The elasticity associated with nearby smaller-sized non-flowering individuals appears to be similarly large relative to other elements.
Let’s now assess the elasticity of \(\lambda\) to matrix elements, comparing the ahistorical to the historically-corrected case.
lathmat2melas <- elasticity3(lathmat2lnmean)
lathmat3melas <- elasticity3(lathmat3lnmean)
writeLines("\nThe largest ahistorical elasticity is associated with element: ")
#>
#> The largest ahistorical elasticity is associated with element:
which(lathmat2melas$elasmats[[7]] == max(lathmat2melas$elasmats[[7]]))
#> [1] 155
writeLines("\nThe largest historically-corrected elasticity is associated with element: ")
#>
#> The largest historically-corrected elasticity is associated with element:
which(lathmat3melas$ah_elasmats[[7]] == max(lathmat3melas$ah_elasmats[[7]]))
#> [1] 155Both analyses agree that \(\lambda\) is most elastic in response to the transition from non-reproductive class 4 to non-reproductive class 5. Historically-corrected elasticities were most strngly different from the ahistorical case particularly for middle-sized non-reproductive classes.
Finally, let’s compare the elasticity of \(\lambda\) in relation to the core life history stages, via a barplot comparison.
elas_put_together <- cbind.data.frame(colSums(lathmat2melas$elasmats[[7]]), colSums(lathmat3melas$ah_elasmats[[7]]))
names(elas_put_together) <- c("ahist", "hist")
rownames(elas_put_together) <- lathmat2melas$stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity of lambda", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topleft", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 12. Ahistorical vs. historically-corrected elasticity of lambda to stage
The barplots show that incorporating individual history leads to greater elasticity values in larger adults, and in flowering adults, than in the ahistorical case. Although the overall pattern looks similar, the two analyses nonetheless suggest a different importance to young vs. old, small vs. large, and to flowering vs. non-flowering.
Analysis 3: Integral projection models (IPMs)
In this analysis, we will build historical and ahistorical integral projection models (IPMs). An IPM is essentially a form of function-based matrix, but one in which transitions are modeled on a continuous state variable rather than discrete stages (Ellner and Rees 2006). Size is often used as this state variable. In practice, the matrix requires discrete size classes, and so size is modeled as Gaussian and the size distribution is broken up into many fine-scale, equally sized classes or size bins. Although the number of these bins varies from analysis to analysis, package lefko3 uses a default of 100 (this can be changed as an option). IPMs are often complex, meaning that they include some life history stages that fall outside of the size classifications developed for the matrices, such as dormant seeds or juveniles. Lefko3 can handle all of this.
Step 1. Life history model development
To begin, we need to create a stageframe for this dataset, in which we identify the key stages used in analysis. We will base all of this on Ehrlén (2000), but use a different size classification to allow IPM construction and make all mature stages other than vegetative dormancy reproductive.
In the stageframe code below, we show that we want an IPM by choosing two stages that serve as the size limits for IPM size classification. These two size classes should have exactly the same characteristics in the stageframe other than size. By choosing these two, we can skip adding and describing the many size classes that will fall between these limits - sf_create() will create all of these for us. We mark these limits in the vector that we load into the stagenames option using the string ipm. Package lefko3 will then rename all IPM size classes according to its own conventions. The default number of size classes is 100 bins. We can alter this if we wish using the ipmbins option.
rm(list=ls(all=TRUE))
data(lathyrus)
sizevector <- c(0, 100, 0, 1, 7100)
stagevector <- c("Sd", "Sdl", "Dorm", "ipm", "ipm")
repvector <- c(0, 0, 0, 1, 1)
obsvector <- c(0, 1, 0, 1, 1)
matvector <- c(0, 0, 1, 1, 1)
immvector <- c(1, 1, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0)
indataset <- c(0, 1, 1, 1, 1)
binvec <- c(0, 100, 0.5, 1, 1)
lathframeipm <- sf_create(sizes = sizevector, stagenames = stagevector, repstatus = repvector,
obsstatus = obsvector, matstatus = matvector, immstatus = immvector,
indataset = indataset, binhalfwidth = binvec, propstatus = propvector,
ipmbins = 100, roundsize = 3)We will also add some descriptive comments to this stageframe so that we know what each of these stages is, and then look at the first 6 entries and the dimenstions of the stageframe.
lathframeipm$comments <- c("Dormant seed", "Seedling", "Dormant", rep("ipm stage", 100))
head(lathframeipm)
#> stagenames size repstatus obsstatus propstatus immstatus matstatus indataset binhalfwidth_raw min_age max_age sizebin_min
#> 1 Sd 0.000 0 0 1 1 0 0 0.000 NA NA 0.00
#> 2 Sdl 100.000 0 1 0 1 0 1 100.000 NA NA 0.00
#> 3 Dorm 0.000 0 0 0 0 1 1 0.500 NA NA -0.50
#> 4 sz36.495 rp1 mt1 ob1 36.495 1 1 0 0 1 1 35.495 NA NA 1.00
#> 5 sz107.485 rp1 mt1 ob1 107.485 1 1 0 0 1 1 35.495 NA NA 71.99
#> 6 sz178.475 rp1 mt1 ob1 178.475 1 1 0 0 1 1 35.495 NA NA 142.98
#> sizebin_max sizebin_center sizebin_width comments
#> 1 0.00 0.000 0.00 Dormant seed
#> 2 200.00 100.000 200.00 Seedling
#> 3 0.50 0.000 1.00 Dormant
#> 4 71.99 36.495 70.99 ipm stage
#> 5 142.98 107.485 70.99 ipm stage
#> 6 213.97 178.475 70.99 ipm stage
This stageframe shows has 103 stages (we can see this by typing dim(lathframeipm)). The IPM portion technically starts with the fourth stage and keeps going until the 103rd stage. Stage names within this range are concatenations of the central size (designated with sz), reproductive status (designated with rp), maturity status (designated with mt), and observation status (designated with ob). The first three stages, which fall outside of the IPM classification, are left unaltered.
Step 2a. Dataset organization
To work with this dataset, we first need to format the data into ‘vertical’ format, in which each row corresponds to the state of a single individual in two (if ahistorical) or three (if historical) consecutive time intervals. Because this is an IPM, we will need to estimate linear models of vital rates, and that will require that NAs in size and fecundity are avoided in key terms used in estimation. For this purpose, we will set NAas0 = TRUE. We will also set NRasRep = TRUE because all adult stages are assumed to be reproductive, and there are mature individuals in the dataset that do not reproduce but need to be included in reproductive stages (setting this option to TRUE makes sure that the reproductive status of non-reproductive individuals is set to 1, although the actual fecundity is not altered). Finally, we will ignore patches this time and estimate matrices only for the full population, because we also wish to estimate historical IPMs, which will be quite large.
lathvertipm <- verticalize3(lathyrus, noyears = 4, firstyear = 1988,
individcol = "GENET", blocksize = 9, juvcol = "Seedling1988",
size1col = "Volume88", repstr1col = "FCODE88",
fec1col = "Intactseed88", dead1col = "Dead1988",
nonobs1col = "Dormant1988", stageassign = lathframeipm,
stagesize = "sizea", censorcol = "Missing1988",censorkeep = NA,
censor = TRUE, NAas0 = TRUE, NRasRep = TRUE)Voila! We encourage users to explore the created dataset, which now includes 2,527 historical transitions (rows) and 42 variables (columns). Now we move on to create the extra bits of information needed for matrix estimation.
Step 2b: Develop supplemental information for matrix estimation
Now we will provide some given transitions. These are the same overwrite objects as used before.
lathover2 <- overwrite(stage3 = c("Sd", "Sdl"), stage2 = c("Sd", "Sd"),
givenrate = c(0.345, 0.054))
lathover3 <- overwrite(stage3 = c("Sd", "Sd", "Sdl"), stage2 = c("Sd", "Sd", "Sd"),
stage1 = c("Sd", "rep", "rep"), givenrate = c(0.345, 0.345, 0.054))
We will also create a reproductive matrix, which describes not only which stages are reproductive, but which stages they lead to the reproduction of, and at what level. This matrix is composed mostly of 0s, but fecundity is noted as non-zero entries equal to a scalar multiplier to the full fecundity estimated by R. This matrix has rows and columns equal to the number of stages described in the stageframe for this dataset, and the rows and columns refer to these stages in the same order as in the stageframe. In many ways, it looks like a nearly empty population matrix, but notes the per-individual mean modifiers on fecundity for each stage that actually reproduces. Here, we first create a 0 matrix with dimensions equal to the number of rows in lathframeipm. Then we modify elements corresponding to fecundity by the expected mean seed dormancy probability (row 1), and by the germination rate (row 2).
Let’s now use the same rounding procedure as in Analysis 2 for fecundity in our vertical dataset, since we know that fecundity is all integers except for 6 data points that were estimated differently from the rest. Then we will take a peek at the degree to which 0s exist in fecundity.
lathvertipm$feca2 <- round(lathvertipm$feca2)
lathvertipm$feca1 <- round(lathvertipm$feca1)
lathvertipm$feca3 <- round(lathvertipm$feca3)
length(subset(lathvertipm, repstatus2 == 1)$feca2)
#> [1] 599
length(which(subset(lathvertipm, repstatus2 == 1)$feca2 > 0))
#> [1] 265Over half of reproductive individuals do not actually reproduce in this example. Under these circumstances, we will not be able to use the Poisson or negative binomial distribution to model fecundity, and so must rely on the Gaussian.
Step 3. Tests of history, and vital rate modeling
Integral projection models (IPMs) require functions of vital rates to populate them. Here, we will develop these functions as linear models using modelsearch(). This looks similar to the modelsearch call in the last example, although we will not include models of reproductive status because we assume that all adults are reproductive (though perhaps not successfully so). First we will create the historical models in order to assess whether history is a significant influence on vital rates.
lathmodels3ipm <- modelsearch(lathvertipm, historical = TRUE, approach = "lme4", suite = "size",
vitalrates = c("surv", "obs", "size", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "gaussian",
indiv = "individ", year = "year2", year.as.random = TRUE,
show.model.tables = TRUE, quiet = TRUE)
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0189452 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: very large eigenvalue
#> - Rescale variables?;Model is nearly unidentifiable: large eigenvalue ratio
#> - Rescale variables?
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
summary(lathmodels3ipm)
#> This LefkoMod object includes 7 linear models.
#> Best-fit model criterion used: AICc&k
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea1 + (1 | individ) + (1 | year2)
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 910.3733 933.2409 -451.1866 902.3733 2242
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.2571
#> year2 (Intercept) 0.6149
#> Number of obs: 2246, groups: individ, 257; year2, 3
#> Fixed Effects:
#> (Intercept) sizea1
#> 2.589549 0.001786
#> convergence code 0; 0 optimizer warnings; 3 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: obs.data
#> AIC BIC logLik deviance df.resid
#> 1353.5346 1370.5136 -673.7673 1347.5346 2118
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.01967
#> year2 (Intercept) 0.00000
#> Number of obs: 2121, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.235
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ sizea1 + sizea2 + (1 | individ) + (1 | year2) + sizea1:sizea2
#> Data: size.data
#> REML criterion at convergence: 29132.25
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.0
#> year2 (Intercept) 247.2
#> Residual 480.4
#> Number of obs: 1916, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept) sizea1 sizea2 sizea1:sizea2
#> 8.998e+01 3.119e-01 5.954e-01 -9.417e-05
#> fit warnings:
#> Some predictor variables are on very different scales: consider rescaling
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Reproductive status model:
#> [1] 1
#>
#> ────────────────────────────────────────
#>
#> Fecundity model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: feca2 ~ sizea2 + (1 | individ) + (1 | year2)
#> Data: fec.data
#> REML criterion at convergence: 12785.87
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.7022
#> year2 (Intercept) 0.9860
#> Residual 4.0934
#> Number of obs: 2246, groups: individ, 257; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> -0.521168 0.003196
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Juvenile survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ (1 | individ) + (1 | year2)
#> Data: juvsurv.data
#> AIC BIC logLik deviance df.resid
#> 323.6696 334.5847 -158.8348 317.6696 278
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0
#> year2 (Intercept) 0
#> Number of obs: 281, groups: individ, 187; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 1.084
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: juvobs.data
#> AIC BIC logLik deviance df.resid
#> 91.4924 101.5338 -42.7462 85.4924 207
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 16.009
#> year2 (Intercept) 1.221
#> Number of obs: 210, groups: individ, 154; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 10.39
#>
#> ────────────────────────────────────────
#>
#> Juvenile size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ (1 | individ) + (1 | year2)
#> Data: juvsize.data
#> REML criterion at convergence: 1303.003
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.989
#> year2 (Intercept) 1.182
#> Residual 6.997
#> Number of obs: 193, groups: individ, 144; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 11.02
#>
#> ────────────────────────────────────────
#>
#> Juvenile reproduction model:
#> [1] 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Number of models in survival table:4
#>
#> Number of models in observation table:5
#>
#> Number of models in size table:5
#>
#> Number of models in reproduction status table: 1
#>
#> Number of models in fecundity table:5
#>
#> Number of models in juvenile survival table:1
#>
#> Number of models in juvenile observation table:1
#>
#> Number of models in juvenile size table:1
#>
#> Number of models in juvenile reproduction table: 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> General model parameter names (column 1), and specific names used in these models (column 2):
#> mainparams modelparams
#> 1 year2 year2
#> 2 individ individ
#> 3 patch <NA>
#> 4 surv3 alive3
#> 5 obs3 obsstatus3
#> 6 size3 sizea3
#> 7 repst3 repstatus3
#> 8 fec3 feca3
#> 9 fec2 feca2
#> 10 size2 sizea2
#> 11 size1 sizea1
#> 12 repst2 repstatus2
#> 13 repst1 repstatus1
#> 14 age <NA>
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLWe see here, as before, that size in time t-1 exerts an influence on some vital rates, including survival to time t+1 and size in time t+1. So, the historical IPM is the correct choice here. However, we will also create an ahistorical IPM. For that purpose, we will create the ahistorical linear model set.
lathmodels2ipm <- modelsearch(lathvertipm, historical = FALSE, approach = "lme4", suite = "size",
vitalrates = c("surv", "obs", "size", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "gaussian",
indiv = "individ", year = "year2", year.as.random = TRUE,
show.model.tables = TRUE, quiet = TRUE)
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
#> boundary (singular) fit: see ?isSingular
summary(lathmodels2ipm)
#> This LefkoMod object includes 7 linear models.
#> Best-fit model criterion used: AICc&k
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ sizea2 + (1 | individ) + (1 | year2)
#> Data: surv.data
#> AIC BIC logLik deviance df.resid
#> 933.6973 956.5649 -462.8486 925.6973 2242
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.5305
#> year2 (Intercept) 0.0000
#> Number of obs: 2246, groups: individ, 257; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 2.503007 0.001221
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: obs.data
#> AIC BIC logLik deviance df.resid
#> 1353.5346 1370.5136 -673.7673 1347.5346 2118
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.01967
#> year2 (Intercept) 0.00000
#> Number of obs: 2121, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 2.235
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ sizea2 + (1 | individ) + (1 | year2)
#> Data: size.data
#> REML criterion at convergence: 29294.15
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.0
#> year2 (Intercept) 210.9
#> Residual 504.6
#> Number of obs: 1916, groups: individ, 254; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> 164.0695 0.6211
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Reproductive status model:
#> [1] 1
#>
#> ────────────────────────────────────────
#>
#> Fecundity model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: feca2 ~ sizea2 + (1 | individ) + (1 | year2)
#> Data: fec.data
#> REML criterion at convergence: 12785.87
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.7022
#> year2 (Intercept) 0.9860
#> Residual 4.0934
#> Number of obs: 2246, groups: individ, 257; year2, 3
#> Fixed Effects:
#> (Intercept) sizea2
#> -0.521168 0.003196
#>
#> ────────────────────────────────────────────────────────────────────────────────
#> Juvenile survival model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: alive3 ~ (1 | individ) + (1 | year2)
#> Data: juvsurv.data
#> AIC BIC logLik deviance df.resid
#> 323.6696 334.5847 -158.8348 317.6696 278
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0
#> year2 (Intercept) 0
#> Number of obs: 281, groups: individ, 187; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 1.084
#> convergence code 0; 0 optimizer warnings; 1 lme4 warnings
#>
#> ────────────────────────────────────────
#>
#> Juvenile observation model:
#> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
#> Family: binomial ( logit )
#> Formula: obsstatus3 ~ (1 | individ) + (1 | year2)
#> Data: juvobs.data
#> AIC BIC logLik deviance df.resid
#> 91.4924 101.5338 -42.7462 85.4924 207
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 16.009
#> year2 (Intercept) 1.221
#> Number of obs: 210, groups: individ, 154; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 10.39
#>
#> ────────────────────────────────────────
#>
#> Juvenile size model:
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: sizea3 ~ (1 | individ) + (1 | year2)
#> Data: juvsize.data
#> REML criterion at convergence: 1303.003
#> Random effects:
#> Groups Name Std.Dev.
#> individ (Intercept) 0.989
#> year2 (Intercept) 1.182
#> Residual 6.997
#> Number of obs: 193, groups: individ, 144; year2, 3
#> Fixed Effects:
#> (Intercept)
#> 11.02
#>
#> ────────────────────────────────────────
#>
#> Juvenile reproduction model:
#> [1] 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Number of models in survival table:2
#>
#> Number of models in observation table:2
#>
#> Number of models in size table:2
#>
#> Number of models in reproduction status table: 1
#>
#> Number of models in fecundity table:2
#>
#> Number of models in juvenile survival table:1
#>
#> Number of models in juvenile observation table:1
#>
#> Number of models in juvenile size table:1
#>
#> Number of models in juvenile reproduction table: 1
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> General model parameter names (column 1), and specific names used in these models (column 2):
#> mainparams modelparams
#> 1 year2 year2
#> 2 individ individ
#> 3 patch <NA>
#> 4 surv3 alive3
#> 5 obs3 obsstatus3
#> 6 size3 sizea3
#> 7 repst3 repstatus3
#> 8 fec3 feca3
#> 9 fec2 feca2
#> 10 size2 sizea2
#> 11 size1 <NA>
#> 12 repst2 repstatus2
#> 13 repst1 <NA>
#> 14 age <NA>
#>
#>
#> ────────────────────────────────────────────────────────────────────────────────
#>
#> Quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLWe note some strong similarities here, although obviously size in time t-1 is no longer present. Let’s move on now to the matrices themselves.
Step 4. IPM estimation
We will now create the historical suite of matrices covering the years of study. Note that we will use the negfec = TRUE option. This option prevents fecundity estimates added to the matrix to be negative - a situation that can happen when fecundity is treated as Gaussian. Also, be aware that the output matrices will be extremely large - large enough that some computers might have difficulty with them. If you encounter an error message telling you that you hae run out of memory, then please try this on a more powerful computer.
lathmat3ipm <- flefko3(stageframe = lathframeipm, repmatrix = lathrepmipm,
modelsuite = lathmodels3ipm, overwrite = lathover3,
data = lathvertipm, year.as.random = FALSE,
patch.as.random = FALSE, reduce = FALSE, negfec = TRUE)
summary(lathmat3ipm)
#>
#> This historical lefkoMat object contains 3 matrices.
#>
#> Each matrix is a square matrix with 10609 rows and columns, and a total of 112550881 elements.
#> A total of 3122124 survival transitions were estimated, with 1040708 per matrix.
#> A total of 59946 fecundity transitions were estimated, with 19982 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLThese are giant matrices. With 10,609 rows and columns, there are a total of 112,550,881 elements per matrix. But they are also amazingly sparse - with 1,060,690 elements estimated, 0.9% of elements per matrix are non-zero.
Let’s now build the historical IPMs.
lathmat2ipm <- flefko2(stageframe = lathframeipm, repmatrix = lathrepmipm,
modelsuite = lathmodels2ipm, overwrite = lathover2,
data = lathvertipm, year.as.random = FALSE,
patch.as.random = FALSE, reduce = FALSE, negfec = TRUE)
summary(lathmat2ipm)
#>
#> This ahistorical lefkoMat object contains 3 matrices.
#>
#> Each matrix is a square matrix with 103 rows and columns, and a total of 10609 elements.
#> A total of 30624 survival transitions were estimated, with 10208 per matrix.
#> A total of 582 fecundity transitions were estimated, with 194 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULL
The ahistorical IPMs are certainly smaller than the historical IPMs, but are nonetheless huge in comparison to the matrices estimated in previous analyses. Although huge, these matrices are not sparse - 10,402 elements out of 10,609 per matrix are estimated (98.0%). Fortunately, we can assume that neither MPM is overparameterized, since ultimately the elements in an IPM and any other function-based matrix reflect the statistical power of the underlying vital rate models. In lefko3, the vital rate models used are the best-fit models from exhaustive model selection, and so are already the most parsimonious from within the suite of tested models.
Let’s take a look at the top-left corner of the first ahistorical matrix (the matrix is too huge to inspect in full here).
print(lathmat2ipm$A[[1]][1:25,1:5], digits = 3)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.345 0.00e+00 0.00000 0.00000 0.00000
#> [2,] 0.054 0.00e+00 0.00000 0.00000 0.00000
#> [3,] 0.000 4.82e-05 0.08933 0.08962 0.09017
#> [4,] 0.000 4.23e-03 0.04247 0.04173 0.04007
#> [5,] 0.000 2.01e-41 0.04476 0.04426 0.04303
#> [6,] 0.000 1.91e-124 0.04625 0.04602 0.04530
#> [7,] 0.000 3.64e-252 0.04685 0.04692 0.04675
#> [8,] 0.000 0.00e+00 0.04653 0.04689 0.04730
#> [9,] 0.000 0.00e+00 0.04531 0.04595 0.04692
#> [10,] 0.000 0.00e+00 0.04325 0.04414 0.04564
#> [11,] 0.000 0.00e+00 0.04048 0.04157 0.04351
#> [12,] 0.000 0.00e+00 0.03715 0.03839 0.04068
#> [13,] 0.000 0.00e+00 0.03342 0.03475 0.03728
#> [14,] 0.000 0.00e+00 0.02947 0.03085 0.03350
#> [15,] 0.000 0.00e+00 0.02548 0.02684 0.02951
#> [16,] 0.000 0.00e+00 0.02160 0.02290 0.02549
#> [17,] 0.000 0.00e+00 0.01796 0.01915 0.02158
#> [18,] 0.000 0.00e+00 0.01463 0.01571 0.01792
#> [19,] 0.000 0.00e+00 0.01169 0.01263 0.01458
#> [20,] 0.000 0.00e+00 0.00915 0.00995 0.01163
#> [21,] 0.000 0.00e+00 0.00703 0.00769 0.00910
#> [22,] 0.000 0.00e+00 0.00529 0.00583 0.00698
#> [23,] 0.000 0.00e+00 0.00391 0.00433 0.00525
#> [24,] 0.000 0.00e+00 0.00283 0.00315 0.00387
#> [25,] 0.000 0.00e+00 0.00201 0.00225 0.00280This matrix is very large, of course, so is difficult to read properly. We can get another handle on quality control by checking the column sums of the first U matrix, to make sure that all column sums look like survival probabilities.
summary(colSums(lathmat2ipm$U[[1]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.004278 0.983811 0.998781 0.954126 0.999866 0.999985Everything looks OK, with stage survival probabilities within the realm of possibility.
Let’s now repeat with the historical matrices. First the top corner of the first historical matrix.
print(lathmat3ipm$A[[1]][1:25,1:10], digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 0.345 0 0 0 0 0 0 0 0 0
#> [2,] 0.000 0 0 0 0 0 0 0 0 0
#> [3,] 0.000 0 0 0 0 0 0 0 0 0
#> [4,] 0.000 0 0 0 0 0 0 0 0 0
#> [5,] 0.000 0 0 0 0 0 0 0 0 0
#> [6,] 0.000 0 0 0 0 0 0 0 0 0
#> [7,] 0.000 0 0 0 0 0 0 0 0 0
#> [8,] 0.000 0 0 0 0 0 0 0 0 0
#> [9,] 0.000 0 0 0 0 0 0 0 0 0
#> [10,] 0.000 0 0 0 0 0 0 0 0 0
#> [11,] 0.000 0 0 0 0 0 0 0 0 0
#> [12,] 0.000 0 0 0 0 0 0 0 0 0
#> [13,] 0.000 0 0 0 0 0 0 0 0 0
#> [14,] 0.000 0 0 0 0 0 0 0 0 0
#> [15,] 0.000 0 0 0 0 0 0 0 0 0
#> [16,] 0.000 0 0 0 0 0 0 0 0 0
#> [17,] 0.000 0 0 0 0 0 0 0 0 0
#> [18,] 0.000 0 0 0 0 0 0 0 0 0
#> [19,] 0.000 0 0 0 0 0 0 0 0 0
#> [20,] 0.000 0 0 0 0 0 0 0 0 0
#> [21,] 0.000 0 0 0 0 0 0 0 0 0
#> [22,] 0.000 0 0 0 0 0 0 0 0 0
#> [23,] 0.000 0 0 0 0 0 0 0 0 0
#> [24,] 0.000 0 0 0 0 0 0 0 0 0
#> [25,] 0.000 0 0 0 0 0 0 0 0 0The sparseness of the matrix means that the vast majority of it will be composed of 0s. Let’s take a look at a summary of the column sums of the first survival-transition matrix.
summary(colSums(lathmat3ipm$U[[1]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.0000 0.9963 0.9999 0.9690 1.0000 1.0000These numbers also all look fine.
Now let’s estimate the mean IPM matrices. This code will estimate 1 mean matrix each, because we did not separate patches in the data reorganization and vital rate modeling.
lath2ipmmean <- lmean(lathmat2ipm)
summary(lath2ipmmean)
#>
#> This ahistorical lefkoMat object contains 1 matrices.
#>
#> Each matrix is a square matrix with 103 rows and columns, and a total of 10609 elements.
#> A total of 10208 survival transitions were estimated, with 10208 per matrix.
#> A total of 200 fecundity transitions were estimated, with 200 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULL
lath3ipmmean <- lmean(lathmat3ipm)
summary(lath3ipmmean)
#>
#> This historical lefkoMat object contains 1 matrices.
#>
#> Each matrix is a square matrix with 10609 rows and columns, and a total of 112550881 elements.
#> A total of 1040708 survival transitions were estimated, with 1040708 per matrix.
#> A total of 20600 fecundity transitions were estimated, with 20600 per matrix.
#>
#> Vital rate modeling quality control:
#>
#> Survival estimated with 257 individuals and 2246 individual transitions.
#> Observation estimated with 254 individuals and 2121 individual transitions.
#> Size estimated with 254 individuals and 1916 individual transitions.
#> Reproduction probability not estimated.
#> Fecundity estimated with 257 individuals and 2246 individual transitions.
#> Juvenile survival estimated with 187 individuals and 281 individual transitions.
#> Juvenile observation estimated with 154 individuals and 210 individual transitions.
#> Juvenile size estimated with 144 individuals and 193 individual transitions.
#> Juvenile reproduction probability not estimated.
#> NULLAs a final check, let’s take a look at the column sums of the grand mean survival-transition matrix from each case.
writeLines("\nAhistorical matrix stage survival distribution: ")
#>
#> Ahistorical matrix stage survival distribution:
summary(colSums(lath2ipmmean$U[[1]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.004347 0.976516 0.998775 0.945200 0.999866 0.999985
writeLines("\nHistorical matrix stage-pair survival distribution: ")
#>
#> Historical matrix stage-pair survival distribution:
summary(colSums(lath3ipmmean$U[[1]]))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.0000 0.9887 0.9994 0.9604 1.0000 1.0000All looks fine!
Step 5. MPM analysis
Now let’s estimate the deterministic population growth rates and plot them.
ipm2lambda <- lambda3(lathmat2ipm)
ipm3lambda <- lambda3(lathmat3ipm)
meanlambda2 <- lambda3(lath2ipmmean)
meanlambda3 <- lambda3(lath3ipmmean)
plot(lambda ~ year2, data = ipm2lambda, xlab = "Year", ylab = "Lambda", ylim = c(0.65, 1.00),
type = "l", lwd = 2, bty = "n")
lines(lambda ~ year2, data = ipm3lambda, lwd = 2, lty = 2, col = "red")
legend("bottomleft", c("ahistorical", "historical"), lty = c(1, 2), col = c("black", "red"), lwd = 2, bty = "n")
Figure 13. Ahistorical vs. historical lambda
Ahistorical estimates of \(\lambda\) are lower than historical estimates, and the historical \(\lambda\) values are more in line with estimates from the previous analyses.
Now let’s look at the stable stage distribution. We will only look at the summary and first 6 entries, given the size of the output.
ipm2ss <- stablestage3(lath2ipmmean)
ipm3ss <- stablestage3(lath3ipmmean)
ss_put_together <- cbind.data.frame(ipm2ss$ss_prop, ipm3ss$ahist$ss_prop)
names(ss_put_together) <- c("ahist", "hist")
rownames(ss_put_together) <- ipm2ss$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topright", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 14. Ahistorical vs. historically-corrected stable stage distribution
Both plot show the population dominated by the first stage, which is the dormant seed stage. The importance of the seed bank to the population is quite clear in this analysis! However, the historical analysis suggests a stronger weighting of larger adults, with the distribution moving rightward slightly.
Next, the reproductive values.
ipm2rv <- repvalue3(lath2ipmmean)
ipm3rv <- repvalue3(lath3ipmmean)
rv_put_together <- cbind.data.frame(ipm2rv$rep_value, ipm3rv$ahist$rep_value)
names(rv_put_together) <- c("ahist", "hist")
rv_put_together$ahist <- rv_put_together$ahist / max(rv_put_together$ahist)
rv_put_together$hist <- rv_put_together$hist / max(rv_put_together$hist)
rownames(rv_put_together) <- ipm2rv$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topleft", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 15. Ahistorical vs. historically-corrected reproductive values
A quick scan through these values shows that the highest reproductive values in the ahistorical analysis are for the largest adults, all the way at the bottom of the data frame output. Since these values have been scaled to the contribution of dormant seed, the reproductive values suggest the important contribution of large adults to the maintenance of the population. However, historically-corrected reproductive values drop off once adults get mildly large, with the largest value associated with plants with a leaf volume of 5076 (the maximum is >7000).
Given the size of these matrices and the difficulty of working with them, we will skip sensitivity analysis here and move on to elasticity analysis.
lath2ipmelas <- elasticity3(lath2ipmmean)
lath3ipmelas <- elasticity3(lath3ipmmean)
writeLines("\nThe highest ahistorical elasticity is associated with element: ")
#>
#> The highest ahistorical elasticity is associated with element:
which(lath2ipmelas$elasmats[[1]] == max(lath2ipmelas$elasmats[[1]]))
#> [1] 209
writeLines("\nThe highest historically-corrected elasticity is associated with element: ")
#>
#> The highest historically-corrected elasticity is associated with element:
which(lath3ipmelas$ah_elasmats[[1]] == max(lath3ipmelas$ah_elasmats[[1]]))
#> [1] 209Both analyses agree that \(\lambda\) is most elastic in response to stasis in vegetative dormancy, which is rather different from previous analyses, which suggested that transitions involving small adults or flowering adults had the strongest potential influence on \(\lambda\). Let’s plot the importance of stages from both perspectives.
elas_put_together <- cbind.data.frame(colSums(lath2ipmelas$elasmats[[1]]), colSums(lath3ipmelas$ah_elasmats[[1]]))
names(elas_put_together) <- c("ahist", "hist")
rownames(elas_put_together) <- lath2ipmelas$stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity of lambda", xlab = "Stage",
col = c("black", "red"), bty = "n")
legend("topright", c("ahistorical", "historical"), col = c("black", "red"), pch = 15, bty = "n")
Figure 16. Ahistorical vs. historically-corrected elasticity of lambda to stage
The plot of these distributions shows the strong importance of vegetative dormancy, which is the tallest bar in both plots. However, the distribution of elasticity values in adult stages is shifted to the right in the historically-corrected IPM. Thus, while the ahistorical IPM shows the second highest elasticity of \(\lambda\) to be associated with plants with a leaf volume of 746, historically-corrected analysis suggests that the second highest elasticity is associated with plants with a leaf volume of 960.