CASE STUDIES OF SWEDISH Lathyrus vernus POPULATION
In this vignette, we will use the lathyrus dataset to illustrate the estimation of raw MPMs, with the intention of producing matrices similar to those published in Ehrlén (2000). Please see the other vignettes included in package lefko3, as well as further vignettes posted online on the projects page of the Shefferson lab website. Other vignettes include demonstrations of function-based MPMs and IPMs, as well as age-by-stage MPMs.
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 (Ehrlen and Lehtila 2002). Lathyrus vernus lacks organs for vegetative spread and individuals are well delimited (Ehrlen 2002). One or several erect shoots of up to 40 cm height emerge from a subterranean rhizome in March and April. Flowering occurs about four weeks after shoot emergence. Shoot growth is determinate, and the number of flowers is determined in the previous year (Ehrlen and Van Groenendael 2001). Individuals sometimes do not produce aboveground structures every year, instead remaining 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 (Ehrlen 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 (Ehrlen and Munzbergova 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 (Ehrlen 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 growing 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 strongly 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 (Ehrlen and Eriksson 1996).
OVERALL GOALS AND INITIAL CONSIDERATIONS
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 2.1. 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. It is a good idea to also look at a summary of the dataset.
rm(list=ls(all=TRUE))
library(lefko3)
data(lathyrus)
#summary(lathyrus)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 completely match those used in the dataset. 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 bin widths can be used (but please do not use 0). 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, which gives the stage a size range covering -0.5 to 0.5. The lowest observable stage must then have a size bin going no lower than 0.5 in order to avoid overlap. Additionally, stagenames must include unique names only, and repstatus, obsstatus, matstatus, immstatus, propstatus, and indataset are binomial vectors referring to status as a reproductive stage, status as an observed stage, status as a mature stage, status as an immature stage, status as a propagule stage, and status as a stage occurring within the user-supplied dataset, respectively. The combination of these characteristics must be completely unique for each stage. The final vector, called comments, is a vector of text stage descriptions that we will use simply to remind ourselves what each stage refers to, in the vernacular. You may type lathframe at the prompt afterwards to inspect the result.
sizevector <- c(0, 100, 13, 127, 3730, 3800, 0)
stagevector <- c("Sd", "Sdl", "Tm", "Sm", "La", "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)
comments <- c("Dormant seed", "Seedling", "Tiny vegetative", "Small vegetative",
"Large vegetative", "Flowering", "Vegetatively dormant")
lathframe <- sf_create(sizes = sizevector, stagenames = stagevector,
repstatus = repvector, obsstatus = obsvector, propstatus = propvector,
immstatus = immvector, matstatus = matvector, indataset = indataset,
binhalfwidth = binvec, comments = comments)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 not observable, such as in the case of vegetative dormancy. However, a size of 0 may have other meanings. For example, if the size metric used is a logarithm of the measured size, then observable sizes of 0 and lower may be possible. These situations may impact matrix construction and analysis, particularly when dealing with function-based MPMs, such as IPMs. See the other vignettes for examples dealing with this.
Step 2a. Data reorganization
Next, we need to reorganize the dataset 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. The verticalize3() function handles this, creaing a historical vertical dataset. Note that most of the inputs to this function constitute the names of the first variables coding for particular states. For example, Seedling1988 is the first variable in the dataset coding for status as a juvenile, Volume88 is the first variable coding for the main size metric, and Intactseed88 is the first variable coding for fecundity. The dataset includes a repeating pattern of such variables organized as blocks of 9 variables for each year (noted as blocksize). There are 4 observation times (noted as noyears). We also have a repeated censor variable, the first of which is Missing1988, and we note here that we wish to censor the data and to keep data points with NA values in the censor term. The patchidcol and individcol terms are variables denoting which patch/subpopulation and individual each row of data belongs to, respectively. Finally, stageassign ties the dataset to the correct stageframe.
lathvert <- verticalize3(lathyrus, noyears = 4, firstyear = 1988,
patchidcol = "SUBPLOT", individcol = "GENET", blocksize = 9,
juvcol = "Seedling1988", sizeacol = "Volume88", repstracol = "FCODE88",
fecacol = "Intactseed88", deadacol = "Dead1988", nonobsacol = "Dormant1988",
stageassign = lathframe, stagesize = "sizea", censorcol = "Missing1988",
censorkeep = NA, censor = TRUE)
#summary(lathvert)
It pays to explore the dataset thoroughly, and summaries can help to do that. Typing summary(lathvert) 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 (92.2%). Further, the minimum values for all size variables are 0, suggesting that unobservable stages occur within the dataset (these are instances of vegetative dormancy).
The resulting reorganization has also dramatically changed the dimensions of the dataset. The following lines show the difference between the two datasets. Note that data has been neither lost nor added - it has just been restructured for analysis.
writeLines("Dimensions of original dataset:")
#> Dimensions of original dataset:
dim(lathyrus)
#> [1] 1119 38
writeLines("Dimensions of verticalized dataset:")
#> Dimensions of verticalized dataset:
dim(lathvert)
#> [1] 2527 45Subset 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$stage2 == "Dorm"),c(22:27)])
#> sizea2 repstra2 feca2 fec2added juvgiven2 obsstatus2
#> Min. :0 Min. : NA Min. : NA Min. :0 Min. :0 Min. :0
#> 1st Qu.:0 1st Qu.: NA 1st Qu.: NA 1st Qu.:0 1st Qu.:0 1st Qu.:0
#> Median :0 Median : NA Median : NA Median :0 Median :0 Median :0
#> Mean :0 Mean :NaN Mean :NaN Mean :0 Mean :0 Mean :0
#> 3rd Qu.:0 3rd Qu.: NA 3rd Qu.: NA 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0
#> Max. :0 Max. : NA Max. : NA Max. :0 Max. :0 Max. :0
#> NA's :138 NA's :138
summary(lathvert[which(lathvert$stage2 == "Dorm"),c(28:33)])
#> repstatus2 fecstatus2 matstatus2 alive2 stage2 stage2index
#> Min. :0 Min. :0 Min. :1 Min. :1 Length:138 Min. :7
#> 1st Qu.:0 1st Qu.:0 1st Qu.:1 1st Qu.:1 Class :character 1st Qu.:7
#> Median :0 Median :0 Median :1 Median :1 Mode :character Median :7
#> Mean :0 Mean :0 Mean :1 Mean :1 Mean :7
#> 3rd Qu.:0 3rd Qu.:0 3rd Qu.:1 3rd Qu.:1 3rd Qu.:7
#> Max. :0 Max. :0 Max. :1 Max. :1 Max. :7
writeLines("Dimensions of data subset corresponding to dormant individuals in time t: ")
#> Dimensions of data subset corresponding to dormant individuals in time t:
dim(lathvert[which(lathvert$stage2 == "Dorm"),])
#> [1] 138 45
The fact that vegetatively dormant individuals have been assigned 0 for size does not require us to exercise any extra steps, because the construction of raw matrices depends on the stage designations rather than on the size classifications. In this case, the verticalize3() function has made these assignments, and this can be seen in the summary in the stage2index column, which shows all individuals that are alive and have a size of 0 in time t to be in the 7th stage. Type lathframe and enter at the prompt to check that the 7th stage in the stageframe is really vegetative dormancy.
A further subset summary can teach us how reproduction is handled here. The reproductive status of flowering adults is certainly set as reproductive (see the distribution of values for repstatus2). However, fecundity ranges from 0 to above 60 (see feca2, which codes for fecundity in time t), meaning that some individuals in the reproductive class do not actually produce any offspring. In fact, only 44.2% of flowering plants produced seed in time t. This has happened because our reproductive status variable, FCODE88, notes whether these individuals flowered but not whether they produced seed. Since this plant does not obligately self-reproduce, and even self-reproduction requires pollen deposition by an insect vector, some flowers will yield no seed. This issue does not cause problems for us here, but it may cause difficulties in the creation of function-based matrices if we wish to assume a count-based distribution for fecundity. In any case, it helps to consider whether the definitions used for stages are appropriate, and so whether reproductive status must necessarily be associated with successful reproduction or merely the attempt. Here, we associate it with the latter, but in other vignettes we will reconsider this assumption.
summary(lathvert[which(lathvert$stage2 == "Flo"),c(22:27)])
#> sizea2 repstra2 feca2 fec2added juvgiven2 obsstatus2
#> Min. : 98.4 Min. :1 Min. : 0.000 Min. : 0.000 Min. :0 Min. :1
#> 1st Qu.: 732.5 1st Qu.:1 1st Qu.: 0.000 1st Qu.: 0.000 1st Qu.:0 1st Qu.:1
#> Median :1141.8 Median :1 Median : 0.000 Median : 0.000 Median :0 Median :1
#> Mean :1388.5 Mean :1 Mean : 4.793 Mean : 4.793 Mean :0 Mean :1
#> 3rd Qu.:1758.0 3rd Qu.:1 3rd Qu.: 6.000 3rd Qu.: 6.000 3rd Qu.:0 3rd Qu.:1
#> Max. :7032.0 Max. :1 Max. :66.000 Max. :66.000 Max. :0 Max. :1
summary(lathvert[which(lathvert$stage2 == "Flo"),c(28:33)])
#> repstatus2 fecstatus2 matstatus2 alive2 stage2 stage2index
#> Min. :1 Min. :0.0000 Min. :1 Min. :1 Length:599 Min. :6
#> 1st Qu.:1 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character 1st Qu.:6
#> Median :1 Median :0.0000 Median :1 Median :1 Mode :character Median :6
#> Mean :1 Mean :0.4424 Mean :1 Mean :1 Mean :6
#> 3rd Qu.:1 3rd Qu.:1.0000 3rd Qu.:1 3rd Qu.:1 3rd Qu.:6
#> Max. :1 Max. :1.0000 Max. :1 Max. :1 Max. :6Step 2b. Provide supplemental information for matrix estimation
Now we will create supplemental tables, which provide external data for matrix estimation not included in the main demographic dataset. 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 incorporate both terms as given constants for specific transitions within our matrices, and as multipliers for fecundity, since fecundity will be estimated as the production of seed multiplied by the seed germination rate or the seed dormancy/survival rate.
lathsupp2 <- supplemental(stage3 = c("Sd", "Sdl", "Sd", "Sdl"),
stage2 = c("Sd", "Sd", "rep", "rep"),
givenrate = c(0.345, 0.054, NA, NA),
multiplier = c(NA, NA, 0.345, 0.054),
type = c(1, 1, 3, 3), stageframe = lathframe, historical = FALSE)
#lathsupp2
The supplemental table above will only work with ahistorical MPMs. The next supplemental table will work for historical MPMs. The primary difference is the incorporation of stage in time t-1. 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" and "all" designations in Stage1 - these are shorthand telling R to use all reproductive stages, or all stages in general, respectively, for the time interval.
lathsupp3 <- supplemental(stage3 = c("Sd", "Sd", "Sdl", "Sd", "Sdl"),
stage2 = c("Sd", "Sd", "Sd", "rep", "rep"),
stage1 = c("Sd", "rep", "rep", "all", "all"),
givenrate = c(0.345, 0.345, 0.054, NA, NA),
multiplier = c(NA, NA, NA, 0.345, 0.054),
type = c(1, 1, 1, 3, 3), stageframe = lathframe, historical = TRUE)
#lathsupp3
These two supplemental tables show us that we have survival-transition probabilities (convtype = 1, whereas fecundity rates would be given as convtype = 2 and fecundity multipliers would be convtype = 3), 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 we would enter the stages corresponding to the correct transitions in the eststageX columns, and the givenrate column would be blank.
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 vignettes focused on function-based MPMs and IPMs. We provide only a barebones description here, focusing 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), which are the default vital rates assumed by the function. Here, we also test observation status (obs), which can serve as a proxy for sprouting probability in cases where plants do not necessarily sprout. This dataset also includes juveniles whose vital rates we wish to estimate. We designate this by setting juvestimate to the correct juvenile stage to focus juvenile vital rate estimation on. Because size also varies in juveniles, we set juvsize = TRUE (this setting defaults to FALSE, in which case size is not tested in juvenile vital rate models). To test history with function modelsearch: 1) use the historical vertical dataset as input, 2) set historical = TRUE, 3) input the relevant vital rates to estimate, 4) 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), 5) set the name of the juvenile stage (if juvenile vital rates are to be estimated and such stages occur in the dataset), 6) set the proper distributions to use for size and fecundity, and 7) note which variables code for individual identity (used to treat identity as a random factor in mixed linear models), patch identity (if multiple patches occur in the dataset and vital rates should be estimated with patch as a factor), and observation time. We also set 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", "repst", "fec"), juvestimate = "Sdl",
sizedist = "gaussian", fecdist = "gaussian", indiv = "individ",
year = "year2", juvsize = TRUE, quiet = TRUE)
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0158034 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: very large eigenvalue
#> - Rescale variables?;Model is nearly unidentifiable: large eigenvalue ratio
#> - Rescale variables?
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> boundary (singular) fit: see ?isSingular
#> Warning: Some predictor variables are on very different scales: consider rescaling
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0171506 (tol =
#> 0.002, component 1)
#> Warning in checkConv(attr(opt, "derivs"), opt$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
#> boundary (singular) fit: see ?isSingular
#histtest
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). Since several vital rates show sizea1 as a term in the best-fit model, particularly adult survival, size, and reproductive status, we see that history has a significant impact on the demography of this population and cannot be ignored.
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 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. Note that we will not create matrices for subpopulations in this case (to include them, add the options patch = "all", patchcol = "patchid" to the input below).
ehrlen2 <- rlefko2(data = lathvert, stageframe = lathframe, year = "all",
stages = c("stage3", "stage2"), supplement = lathsupp2, yearcol = "year2",
indivcol = "individ")
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 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)
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 except the very last year, rather than lumping all years together to produce a single matrix. 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.
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 24.667 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 in the development of vital rate models. 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 we’ll estimate 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"), supplement = lathsupp3,
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 several 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 as well. This sparseness occurs because historical matrices are primarily composed of structural 0s. As a result, 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, giving us matrices with 19 fewer rows and columns, and 82.5 / 900 = 9.2% of elements as potentially non-zero.
ehrlen3red <- rlefko3(data = lathvert, stageframe = lathframe,
year = c(1989, 1990), stages = c("stage3", "stage2", "stage1"),
supplement = lathsupp3, yearcol = "year2", indivcol = "individ",
reduce = TRUE)
#summary(ehrlen3red)Next we will create the element-wise mean ahistorical matrix.
ehrlen2mean <- lmean(ehrlen2)
#ehrlen2mean
Function lmean() creates a lefkoMat object, just as rlefko2() and rlefko3() do, retaining all descriptive information from the original lefkoMat object. The full 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 data frame outlining the definitions and order of historical paired stages (hstages, shown as NA in this case because the matrices are ahistorical), a data frame outlining the actual stages as outlined in the stageframe object used to create these matrices (ahstages), a data frame outlining the definitions and order of the matrices (labels), and two quality control elements used in output for the summary() function (matrixqc and dataqc).
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 this 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.
ehrlen2mean$labels
#> pop patch
#> 1 1 1
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 (contained within the $U element of our mean lefkoMat object).
print(colSums(ehrlen2mean$U[[1]]), digits = 3)
#> [1] 0.399 0.745 0.877 0.937 0.966 0.991 0.667All 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). 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.
Now we will estimate the historical element-wise mean matrix. We will use the reduced matrix to simplify later analyses, and show only the top-left corner of the rather large matrix (a section comprised of the first 20 rows and 8 columns of the 30 x 30 matrix - unhashtag the print statement to see).
ehrlen3mean <- lmean(ehrlen3red)
#print(ehrlen3mean$A[[1]][1:20,1:8], digits = 3)The prevalence of 0s in this matrix is normal because most elements are structural 0s and so cannot equal anything else. This matrix is also a raw matrix, meaning that transitions that do not actually occur in the dataset cannot equal anything other than 0.
To understand the dominance of structural 0s in the historical case, let’s take a look at the hstages object associated with this mean matrix.
ehrlen3mean$hstages
#> stage_id_2 stage_id_1 stage_2 stage_1
#> 1 1 1 Sd Sd
#> 2 2 1 Sdl Sd
#> 3 3 2 Tm Sdl
#> 4 4 2 Sm Sdl
#> 5 7 2 Dorm Sdl
#> 6 3 3 Tm Tm
#> 7 4 3 Sm Tm
#> 8 5 3 La Tm
#> 9 7 3 Dorm Tm
#> 10 3 4 Tm Sm
#> 11 4 4 Sm Sm
#> 12 5 4 La Sm
#> 13 6 4 Flo Sm
#> 14 7 4 Dorm Sm
#> 15 3 5 Tm La
#> 16 4 5 Sm La
#> 17 5 5 La La
#> 18 6 5 Flo La
#> 19 7 5 Dorm La
#> 20 1 6 Sd Flo
#> 21 2 6 Sdl Flo
#> 22 4 6 Sm Flo
#> 23 5 6 La Flo
#> 24 6 6 Flo Flo
#> 25 7 6 Dorm Flo
#> 26 3 7 Tm Dorm
#> 27 4 7 Sm Dorm
#> 28 5 7 La 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 probability estimates for historical stages, which is given as the set of column sums in the survival-transition matrix, as below. As long as all of the numbers are between 0 and 1, then all is well at first glance.
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
Let’s try another approach, looking at some conditional historical matrices. Conditional matrices are matrices showing transitions from stage at time t to time t+1, conditional on all individuals having been in the same stage in time t-1. They are calculated from historical MPMs, and the output below shows all conditional matrices developed from the first $A matrix of ehrlen3mean. The first matrix, for example, shows all transitions involving individuals that had been in the dormant seed stage Sd in time t-1, while the last matrix shows transitions involving individuals that had been vegetatively dormant in time t-1.
ehrlen3condmn <- cond_hmpm(ehrlen3mean)
#ehrlen3condmn$Acond[[1]]Quick scans will show many transitions missing, because each stage has only certain stages that can transition from it, and to which it can transition. In this case, further transitions are missing because the MPM is raw, and some transitions are not parameterized because no individuals made them. Further examination of these conditional matrices may yield interesting biological insights useful in user analyses.
Further fine-scale error-checking is of great help for historical matrices, and may require outputting full or conditional MPMs into spreadsheets. Non-zero elements should only exist where biologically and logically possible, and in the case of raw matrices, where individuals were actually recorded transitioning.
One last technique before we move on to analyses that we can use MPMs in: matrix visualization plots. These plots are quite useful in providing a relatively easy to understand of the “spatial spread” of values throughout a matrix. Package lefko3 includes function image3(), which provides an easy way to make these images. Let’s start off by looking at the ahistorical mean matrix.
image3(ehrlen2mean, used = 1)
Figure 2.2. Image of mean ahistorical matrix
#> NULLThe resulting image shows non-zero elements as red spaces, and zero elements as white spaces. Rows and columns are numbered, and we can see that this matrix is reasonably dense.
Now let’s take a look at a mean historical matrix. The historical mean matrix has many more rows and columns, and has more of both zero and non-zero elements. However, it has become a sparse matrix, and it turns out that increasing the numbers of life stages will increase the sparsity of historical projection matrices.
image3(ehrlen3mean, used = 1)
Figure 2.3. Image of mean historical matrix
#> NULLStep 5. MPM analysis
Package lefko3 includes functions to conduct some analyses of population dynamics. Currently, we cover most of the basic deterministic and stochastic analyses - assessing the population growth rate as \(\lambda\) in the deterministic case and as \(a = \text{log} \lambda _{S}\) in the stochastic case, the stable stage distribution or long-run projected mean stage distribution, and the deterministic and stochastic reproductive value vector for estimated matrices. We will start by estimating the asymptotic population growth rate (\(\lambda\)) and the stochastic population growth rate (\(a = \text{log} \lambda _{S}\)) from the ahistorical MPMs and the the population growth rate associated with the mean matrix from that analysis (we will set the seed for R’s random number generation to make the output reproducible). 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, followed by the deterministic growth rate of the mean matrix, and the stochastic population growth rate (\(a = \text{log} \lambda _{S}\)).
writeLines("Deterministic ahistorical: ")
#> Deterministic ahistorical:
lambda3(ehrlen2)
#> pop patch year2 lambda
#> 1 1 1 1988 0.8952585
#> 2 1 1 1989 0.9235493
#> 3 1 1 1990 1.0096490
writeLines("Deterministic mean ahistorical: ")
#> Deterministic mean ahistorical:
lambda3(ehrlen2mean)
#> pop patch lambda
#> 1 1 1 0.9574162
writeLines("Stochastic ahistorical: ")
#> Stochastic ahistorical:
set.seed(42)
slambda3(ehrlen2)
#> pop patch a var sd se
#> 1 1 1 -0.04490197 0.03154986 0.1776228 0.001776228\(\lambda\) for the mean matrix may seem high 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. Note that there are fewer \(\lambda\) estimates in the historical case, and that mean \(\lambda\) and the stochastic population growth rate (\(a = \text{log} \lambda _{S}\)) 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 II, and Pulliam 2014; Shefferson et al. 2018). Alternatively, these patterns may reflect greater year-to-year variability in allocation to aboveground 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 these estimates of \(\lambda\) and \(a = \text{log} \lambda _{S}\) are likely to be more realistic than the higher estimates in the ahistorical case.
writeLines("Deterministic historical: ")
#> Deterministic historical:
lambda3(ehrlen3red)
#> pop patch lambda
#> 1 1 1 0.8859389
#> 2 1 1 0.9743043
writeLines("Deterministic mean historical: ")
#> Deterministic mean historical:
lambda3(ehrlen3mean)
#> pop patch lambda
#> 1 1 1 0.9045179
writeLines("Stochastic historical: ")
#> Stochastic historical:
set.seed(42)
slambda3(ehrlen3red)
#> pop patch a var sd se
#> 1 1 1 -0.1034958 0.02753828 0.1659466 0.001659466
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 Tm 0.22875637
#> 4 1 4 Sm 0.18625613
#> 5 1 5 La 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 6% 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$hist
This data frame is structured in historical format, and so shows the stable stage distribution of stage pairs. We may wish to see which stage pair dominates, in which case we might look at the row with the maximum ss_prop value.
ehrlen3mss$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 $ahist element of the historical stable stage distribution object. Notice below that the ss_prop column shows values that are a bit different from the ahistorical case, suggesting the influence of individual history.
ehrlen3mss$ahist
#> matrix stage_id stage ss_prop
#> 1 1 1 Sd 0.33462326
#> 2 1 2 Sdl 0.04475617
#> 3 1 3 Tm 0.17891749
#> 4 1 4 Sm 0.19538752
#> 5 1 5 La 0.08122417
#> 6 1 6 Flo 0.11465228
#> 7 1 7 Dorm 0.05043913
To see the impact of history on the stable stage distribution, let’s plot the ahistorical and historically-corrected stable stage distributions together. We will also include the stochastic long-run stage distribution in our output, which is estimated with the same function (stablestage3()) but using the stochastic = TRUE option. This will allow us to see the impact of random temporal variation. We will also set the random seed to make our output reproducible.
ehrlen2mss_s <- stablestage3(ehrlen2, stochastic = TRUE, seed = 42)
ehrlen3mss_s <- stablestage3(ehrlen3red, stochastic = TRUE, seed = 42)
ss_put_together <- cbind.data.frame(ehrlen2mss$ss_prop, ehrlen3mss$ahist$ss_prop,
ehrlen2mss_s$ss_prop, ehrlen3mss_s$ahist$ss_prop)
names(ss_put_together) <- c("d_ahist", "d_hist", "s_ahist", "s_hist")
rownames(ss_put_together) <- ehrlen2mss$stage
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.35), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahistorical", "det historical", "stoc ahistorical", "stoc historical"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Figure 2.4. Deterministic and stochastic ahistorical vs. historically-corrected stable stage distributions
Let’s take the deterministic portion first. Accounting for individual history increased the prevalence of dormant seeds and small adults, but decreased the prevalence of dormant seeds, dormant adults, and tiny and large adults. For example, dormant seeds are now expected to compose 33% of the population (29% in the pure ahistorical case), and small and large adults together compose 28% of the population instead of 26%. Now when we also take in the impact of temporal stochasticity, we can see a reduction in the proportion of dormant seeds and likely large and flowering adults, and an increase in the proportion of tiny adults in the historical output.
Let’s take a look at the reproductive values now, in similar order to the stable stage distribution case. Initially, we will create all sets of reproductive value objects, and then we will move on to plotting them. The structure of these objects is the same as that of the stable stage structure outputs, and, in the case of the ahistorical MPM, the reproductive values are also the same as those produced by popbio’s reproductive.value() function. Because the four vectors are all standardized such that the first non-zero reproductive value is set to 1.0, they are on slightly different scales, and so we will make them comparable for plotting purposes by standardizing them relative to their respective maxima.
ehrlen2mrv <- repvalue3(ehrlen2mean)
ehrlen3mrv <- repvalue3(ehrlen3mean)
ehrlen2mrv_s <- repvalue3(ehrlen2, stochastic = TRUE, seed = 42)
ehrlen3mrv_s <- repvalue3(ehrlen3red, stochastic = TRUE, seed = 42)
rv_put_together <- cbind.data.frame((ehrlen2mrv$rep_value / max(ehrlen2mrv$rep_value)),
(ehrlen3mrv$ahist$rep_value / max(ehrlen3mrv$ahist$rep_value)),
(ehrlen2mrv_s$rep_value / max(ehrlen2mrv_s$rep_value)),
(ehrlen3mrv_s$ahist$rep_value / max(ehrlen3mrv_s$ahist$rep_value)))
names(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- ehrlen2mrv$stage
barplot(t(rv_put_together), beside=T, ylab = "Relative reproductive value",
ylim = c(0, 1.2), xlab = "Stage", col = c("black", "orangered", "grey", "darkred"),
bty = "n")
legend("topright", c("det ahist", "det hist", "stoc ahist", "stoc hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Figure 2.5. Ahistorical vs. historically-corrected reproductive values
Both deterministic and stochastic analyses show that flowering adults have the greatest reproductive value in both ahistorical and historical analysis, while dormant seeds have the least. However, the historical MPM suggests a greater contribution of non-flowering but sprouting adult stages, and lower contributions of dormant seeds, seedlings, and 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 $hist element is the reproductive value of the stage pair. We will plot these reproductive values. Here is the deterministic case.
labels <- apply(as.matrix(ehrlen3mrv$hist[, c("stage_2", "stage_1")]), 1, function(X) {
paste(X[1], X[2])
})
barplot(ehrlen3mrv$hist$rep_value, main = "Historical", xaxt = "n",
ylab = "Reproductive value", xlab = "Stage")
text(cex=0.5, y = -0.18, x = seq(from = 0, to = 1.19*length(ehrlen3mrv$hist$stage_2), by = 1.2),
labels, xpd=TRUE, srt=45)
Figure 2.6. Historical reproductive values
In the bar plot, the labels show the stages in the order of stage in time t followed by stage in time t-1. Examining these values reveals that the largest reproductive value is associated with flowering adults that were previously flowering, while the next largest reproductive value is associated with flowering adults that were previously small, and flowering adults that were previously very large. 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 deterministic sensitivity analysis on the ahistorical mean matrix.
ehrlen2sens <- sensitivity3(ehrlen2mean)
print(ehrlen2sens$ah_sensmats[[1]], digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.0182 0.00284 0.0142 0.0116 0.00479 0.00705 0.00347
#> [2,] 0.2061 0.03225 0.1611 0.1312 0.05434 0.07994 0.03936
#> [3,] 0.2565 0.04016 0.2006 0.1633 0.06766 0.09953 0.04900
#> [4,] 0.4110 0.06432 0.3213 0.2616 0.10838 0.15943 0.07849
#> [5,] 0.5639 0.08826 0.4408 0.3589 0.14871 0.21877 0.10770
#> [6,] 0.7221 0.11302 0.5645 0.4596 0.19043 0.28014 0.13791
#> [7,] 0.3067 0.04801 0.2398 0.1952 0.08089 0.11899 0.05858
writeLines("\nThe highest sensitivity value is: ")
#>
#> The highest sensitivity value is:
max(ehrlen2sens$ah_sensmats[[1]])
#> [1] 0.7220817
writeLines("\nThe highest sensitivity value among biologically plausible elements is: ")
#>
#> The highest sensitivity value among biologically plausible elements is:
max(ehrlen2sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])
#> [1] 0.4596282
writeLines("\nThis occurs in element ")
#>
#> This occurs in element
which(ehrlen2sens$ah_sensmats[[1]] == max(ehrlen2sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))
#> [1] 27The highest sensitivity value is associated with a logical 0 - dormant seeds (stage/column 1) cannot transition to flowering adults (stage/row 6), 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 element 27, which is the transition from small adult (stage/column 4) to flowering adult (stage/row 6).
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$h_sensmats to focus in just on historical sensitivities.
ehrlen3sens <- sensitivity3(ehrlen3mean)
writeLines("\nThe highest sensitivity value among biologically plausible elements: ")
#>
#> The highest sensitivity value among biologically plausible elements:
max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)])
#> [1] 0.2756227
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen3sens$h_sensmats[[1]] == max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)]))
#> [1] 313
writeLines("\nThe highest historically-corrected sensitivity value among biologically plausible elements is: ")
#>
#> The highest historically-corrected sensitivity value among biologically plausible elements is:
max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])
#> [1] 0.4885032
writeLines("\nThis occurs in element ")
#>
#> This occurs in element
which(ehrlen3sens$ah_sensmats[[1]] == max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))
#> [1] 27
The first element produced in this analysis 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. Next, $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. Finally, we have the original $A, $U, and $F matrices used as input.
The maximum biologically plausible sensitivity value in the historical matrix appears to be associated with column 10 (small adult in time t-1 to small adult in time t) and row 13 (small adult in time t to flowering in time t+1). This transition is from small adult in times t-1 and t to flowering adult in t+1. The historically-corrected sensitivity analysis finds the \(\lambda\) is most sensitive to the same element as in the ahistorical MPM. So, once again we see the importance of these two stages, but with greater resolution than was possible in the ahistorical case.
Before moving on, let’s repeat the above exercise with stochastic sensitivities. We will only do the historical case here. Note that we have not yet added a conversion from historical to ahistorical sensitivities in the stochastic case, so we will only show the former.
ehrlen3sens_s <- sensitivity3(ehrlen3red, stochastic = TRUE)
writeLines("\nThe highest stochastic sensitivity value among biologically plausible elements: ")
#>
#> The highest stochastic sensitivity value among biologically plausible elements:
max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)])
#> [1] 0.3286162
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen3sens_s$h_sensmats[[1]] == max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)]))
#> [1] 157The highest stochastic sensitivity is associated with element 157, which is found in the 6th column and 7th row. This is the transition from tiny in times t-1 and t to small in time t+1. So, our results appear to be different in the stochastic case, suggesting an impact of temporal environmental stochasticity.
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$ah_elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.00655 0.00000 0.000000 0.0000 0.00000 0.0116 0.00000
#> [2,] 0.01162 0.00000 0.000000 0.0000 0.00000 0.0206 0.00000
#> [3,] 0.00000 0.02784 0.151678 0.0164 0.00052 0.0000 0.00415
#> [4,] 0.00000 0.00127 0.041102 0.1586 0.02210 0.0167 0.02184
#> [5,] 0.00000 0.00000 0.000604 0.0290 0.04851 0.0532 0.01744
#> [6,] 0.00000 0.00000 0.000000 0.0353 0.06862 0.1673 0.00887
#> [7,] 0.00000 0.00315 0.007180 0.0223 0.00896 0.0107 0.00628
writeLines("\nThe maximum elasticity value: ")
#>
#> The maximum elasticity value:
max(ehrlen2elas$ah_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 (stage 6). 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$ah_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 deterministic elasticity value: ")
#>
#> The highest deterministic 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 the element 156, which is at row 6, column 6. This corresponds to the stasis transition of tiny 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. So, accounting for history has changed our perspective on what contributes most to changes in \(\lambda\).
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. Eyeballing this matrix, it appears that the historically-corrected elasticity matrix supports stasis in the small adult stage as the transition that \(\lambda\) is most elastic to, with stasis as flowering adult a close second. This is a little different from the ahistorical case, which flipped the importance of these two.
print(ehrlen3elas$ah_elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0 0 0.00000 0.0000 0.00000 0.0000 0.00000
#> [2,] 0 0 0.00000 0.0000 0.00000 0.0000 0.00000
#> [3,] 0 0 0.16285 0.0369 0.00013 0.0000 0.00261
#> [4,] 0 0 0.03816 0.1778 0.03268 0.0347 0.02064
#> [5,] 0 0 0.00000 0.0342 0.05415 0.0653 0.01057
#> [6,] 0 0 0.00000 0.0374 0.06812 0.1758 0.00532
#> [7,] 0 0 0.00152 0.0177 0.00906 0.0108 0.00349The historical matrices we used as input are reduced, but the historically-corrected output is in the original dimensions corresponding to the number of stages listed in the stageframe. This makes it relatively easy to compare against the actual ahistorical elasticity matrix, for example by subtracting one matrix from the other to see the differences. We can also sum the columns in this exercise to assess overall shifts in the importance of stages.
print((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]]), digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] -0.00655 0.00000 0.000000 0.00000 0.000000 -0.01162 0.00000
#> [2,] -0.01162 0.00000 0.000000 0.00000 0.000000 -0.02063 0.00000
#> [3,] 0.00000 -0.02784 0.011174 0.02056 -0.000390 0.00000 -0.00154
#> [4,] 0.00000 -0.00127 -0.002939 0.01928 0.010574 0.01801 -0.00120
#> [5,] 0.00000 0.00000 -0.000604 0.00518 0.005642 0.01208 -0.00687
#> [6,] 0.00000 0.00000 0.000000 0.00208 -0.000497 0.00849 -0.00356
#> [7,] 0.00000 -0.00315 -0.005658 -0.00464 0.000100 0.00018 -0.00279
writeLines("\nColumn sums of differences:")
#>
#> Column sums of differences:
colSums((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]]))
#> [1] -0.018169024 -0.032252189 0.001972272 0.042457907 0.015428755 0.006513259 -0.015950980Accounting for history shows \(\lambda\) to be less elastic in response to early-life transitions, transitions from vegetative 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.
Next, we will look at a barplot of the elasticities of life history stages from ahistorical vs. historically-corrected analyses. We will also incorporate stochastic elasticity analysis here, which is likely to show differences given the importance of temporal environmental stochasticity on our analyses so far.
ehrlen2elas_s <- elasticity3(ehrlen2, stochastic = TRUE)
ehrlen3elas_s <- elasticity3(ehrlen3red, stochastic = TRUE)
elas_put_together <- cbind.data.frame(colSums(ehrlen2elas$ah_elasmats[[1]]),
colSums(ehrlen3elas$ah_elasmats[[1]]), colSums(ehrlen2elas_s$ah_elasmats[[1]]),
colSums(ehrlen3elas_s$ah_elasmats[[1]]))
names(elas_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_put_together) <- ehrlen2elas$ah_stages$stage
barplot(t(elas_put_together), beside=T, ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Figure 2.7. Ahistorical vs. historically-corrected deterministic and stochastic elasticity to stages
We see the overall importance of both history and temporal environmental stochasticity here. Ahistorical analyses generally find that population growth rate is more elastic in response to dormant seeds, seedlings, and dormant adults, and less elastic in small adults than in historical analyses. Stochastic analyses suggest that population growth rate is barely elastic in response to dormant seeds and seedlings, and more elastic in response to tiny and small adults.
Let’s now look at the elasticities of different kinds of stages. For this, we will use the summary.lefkoElas() function, which outputs data frames summarizing elasticity sums by the kind of transition. First, we will compare ahistorical against historically-corrected transitions.
ehrlen2elas_sums <- summary(ehrlen2elas)
ehrlen3elas_sums <- summary(ehrlen3elas)
ehrlen2elas_s_sums <- summary(ehrlen2elas_s)
ehrlen3elas_s_sums <- summary(ehrlen3elas_s)
elas_sums_together <- cbind.data.frame(ehrlen2elas_sums$ahist[,2],
ehrlen3elas_sums$ahist[,2], ehrlen2elas_s_sums$ahist[,2],
ehrlen3elas_s_sums$ahist[,2])
names(elas_sums_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_sums_together) <- ehrlen2elas_sums$ahist$category
barplot(t(elas_sums_together), beside=T, ylab = "Elasticity",
xlab = "Transition", col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Figure 2.8. Ahistorical vs. historically-corrected deterministic and stochastic elasticity to transitions
We see similar patterns, though clearly fecundity is less influential on \(\lambda\) in hMPMs than in ahMPMs.Stasis and shrinkage also appear more important in stochastic analysis than in deterministic analysis.
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.