# Lathyrus vernus raw MPMs

This document was built in Markdown in R 4.1.1 and compiled on 06 September 2021. It covers package lefko3 version 3.8.0. Please note that this vignette was written with space considerations in mind. To reduce output size, we have prevented some statements from running if they produce long stretches of output. Examples include most summary() calls. In these cases, we include hashtagged versions of these calls, and ask the user to run these statements without hashtags. We have also written this vignette using the default Ehrlén format as the output format for all historical MPMs. All examples work with deVries format as well.

## 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 (Ehrlén and Lehtilä 2002). L. 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 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 (Ehrln and Van Groenendael 2001). Individuals do not necessarily produce aboveground structures every year, and instead can remain dormant in one or more seasons. L. 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), from 1988 to 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 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 aboveground 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 (Ehrlén 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 monitoring. 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 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 occasions (which we refer to as years here, since there was one main census in each year). The original spreadsheet file 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, 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 occasion t).

Let’s take a look at this dataset. First, we clear memory and load it. We will then look at the dimensions and a summary of the dataset.

rm(list=ls(all=TRUE))
library(lefko3)

data(lathyrus)
dim(lathyrus)
#> [1] 1119   38
#summary(lathyrus)

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. This strictly repeated pattern allows us to manipulate the original dataset quickly and efficiently via lefko3. We should know the number of years used, which is 4 years here (includes all years from and including 1988 to 1991). Ideally, we should also have arranged the columns in the same order for each year, with years in consecutive order with no extra columns between them. Note that this order is not required, provided that we are willing to input all variable names in the correct order when transforming the dataset later.

#### 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. In essence, it links the dataset to the life cycle graph used to model the organism’s life history. 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 we have specified, 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,
#lathframe

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 monitoring occasions. The verticalize3() function handles this, creating a historical vertical dataset. We can use one of two approaches for this task, both using this function. In the most general case, we may input all column names from the original dataset in each input option in the same order (corresponding to monitoring occasion). This allows any format of dataset to be used. However, an easier approach may be used in datasets with strictly repeating sets of columns. In this case, most of the inputs to this function’s options 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 monitoring occasions (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)

The verticalize3() function includes error-checking measures designed to find instances where individual characteristics do not match those assigned to stages in the associated stageframe. In those instances, warning messages will be displayed and the instances will be marked NoMatch in stage1, stage2, or stage3.

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 occasion 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 occasion 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 original dataset and the vertically formatted dataset. 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   45

Subset summaries may shed more light on these data. For example, after subsetting to only cases in which individuals are vegetatively dormant in occasion t and narrowing the summary to only variables corresponding to occasion 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 occasion t: ")
#> Dimensions of data subset corresponding to dormant individuals in occasion t:
dim(lathvert[which(lathvert$stage2 == "Dorm"),]) #> [1] 138 45 The verticalize3() function has made stage assignments for all individuals at each time. This can be seen in the subset summary in the stage2index column, which shows all individuals that are alive and have a size of 0 in occasion t to be in the 7th stage. Type lathframe and enter at the prompt to check that the 7th stage in the stageframe is really vegetative dormancy. A further subset summary can teach us how reproduction is handled here. The reproductive status of flowering adults is set as reproductive in lathframe, and a subset summary of just those individuals flowering in occasion t shows all of those individuals set to reproductive (see the distribution of values for repstatus2 in the output below). However, fecundity ranges from 0 to above 60 (see feca2, which codes for fecundity in occasion t), meaning that some flowering adults do not actually produce any offspring. In fact, only 44.2% of flowering plants produced seed in occasion t. This has happened because our reproductive status variable, FCODE88, notes whether these individuals flowered but not whether they produced seed. Since this plant must be pollinated by an appropriate insect vector, some flowers should yield no seed. This issue does not cause problems in the creation of raw matrices, but it might cause difficulties in the creation of function-based matrices under some conditions. It helps to consider whether the definitions used for stages are appropriate, and so whether reproductive status must necessarily be associated with successful reproduction or merely the attempt. Here, we associate it with the latter, but in other vignettes we will reconsider this assumption. summary(lathvert[which(lathvert$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

#### 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 first create ahistorical MPMs. The functions we will use to build these MPMs include rlefko2() for raw ahistorical MPMs, and rlefko3() for raw historical MPMs.

Ehrlén (2000) shows a mean matrix covering years 1989 and 1990 as occasion 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")
#ehrlen2

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) agestages: a data frame showing the order of age-stage pairs (given if matrices are age-by-stage ahistorical; 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) Input options for function rlefko2() include 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 occasion t). Matrix-estimating functions in lefko3 have a default behavior of creating matrices for each year in the dataset except the final year, rather than lumping all years together to produce a single matrix. However, patches or subpopulations will only be separated if a patch ID variable is provided as input. Package lefko3 includes a great deal of flexibility here to estimate matrices covering all of the populations, patches, and years occurring in a 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. #> #> Survival probability sum check (each matrix represented by column in order): #> [,1] [,2] [,3] #> Min. 0.000 0.399 0.399 #> 1st Qu. 0.576 0.811 0.777 #> Median 0.938 0.921 0.947 #> Mean 0.713 0.841 0.837 #> 3rd Qu. 0.955 0.973 0.980 #> Max. 0.996 1.000 1.000 #> NULL We start off learning that 3 full A matrices were estimated (as well as their U and F decompositions), and that they are 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 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 matrix quality. For example, a raw ahistorical matrix with many zero elements suggests that the dataset may be too small for the developed stageframe. Finally, the last section shows us summaries of the column sums form the survival-transition (U) matrices in this lefkoMat object. These column sums correspond to the survival probabilities of the different life stages, and so the summaries must show numbers ranging from 0.0 to 1.0. All of these values are within the realm of possibility for survival 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 error-checking can be accomplished by checking specific elements within each matrix to see if they match expectation. Now we’ll estimate historical matrices. Because of the size of these matrices, we will only show the lefkoMat summary. Note that we will only create Ehrlén-format matrices - users wishing to create deVries-format hMPMs should add format = "deVries" to the input options (the resulting matrices will be bigger, but contain the same number of estimated elements). 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 158 survival transitions were estimated, with 79 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. #> #> Survival probability sum check (each matrix represented by column in order): #> [,1] [,2] #> Min. 0.000 0.000 #> 1st Qu. 0.000 0.000 #> Median 0.000 0.679 #> Mean 0.411 0.518 #> 3rd Qu. 0.980 1.000 #> Max. 1.000 1.000 #> NULL The summary output shows several differences from the ahistorical case. First, there is one less matrix estimated in the historical case than in the ahistorical case, because raw historical matrices require three consecutive occasions to estimate each transition instead of two. Second, these matrices are larger than ahistorical matrices, with the numbers of rows and columns generally equaling the number of ahistorical rows and columns squared (although deVries-format increases the dimensions through the addition of a prior stage for newborns, and in both Ehrlén and deVries formats the numbers of rows and columns may be reduced under some circumstances as shown in the next block). 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 results from historical matrices being composed primarily of structural 0s. As a result, the historical matrices in this example have non-zero entries in 86 / 2401 = 3.6% 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. This can be done by setting reduce = TRUE, which tells rlefko3() to eliminate stage pairs in which both column and row are zero vectors. Here, we now have matrices with 19 fewer rows and columns, and 86 / 900 = 9.6% 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 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 age-by-stage combinations used (agestages, shown as NA because these are not age-by-stage matrices), 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 numbering 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.

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, corresponding to the rows and columns of the historical matrices. The pairs are interpreted so that matrix columns represent stages in occasions t-1 and t, and the rows represent stages in occasions 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 occasion 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 occasions t-1 and t (column pair), to dormant seed at occasions t and t+1 (row pair) - the occasion t stages match, and so this element is possible. However, element [1, 2] represents the transition probability from seedling in occasion t-1 and very small adult in occasion t (column pair), to dormant seed in occasion t and in occasion 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.lefkoMat() function to assess the key quality control characteristics of the mean hMPM, such as the distribution of survival probability estimates for historical stages.

summary(ehrlen3mean)
#>
#> This historical lefkoMat object contains 1 matrix.
#>
#> Each matrix is a square matrix with 30 rows and columns, and a total of 900 elements.
#> A total of 98 survival transitions were estimated, with 98 per matrix.
#> A total of 8 fecundity transitions were estimated, with 8 per matrix.
#>
#> Survival probability sum check (each matrix represented by column in order):
#>          [,1]
#> Min.    0.000
#> 1st Qu. 0.500
#> Median  0.952
#> Mean    0.759
#> 3rd Qu. 1.000
#> Max.    1.000
#> NULL

Let’s try another approach, looking at some conditional historical matrices. Conditional matrices are matrices showing transitions from stage at occasion t to occasion t+1, conditional on all individuals having been in the same stage in occasion t-1. They are calculated from historical MPMs, and the output below shows all conditional matrices developed from the first $A matrix of ehrlen3mean. The first matrix, for example, shows all transitions involving individuals that had been in the dormant seed stage Sd in occasion t-1, while the last matrix shows transitions involving individuals that had been vegetatively dormant in occasion t-1. ehrlen3condmn <- cond_hmpm(ehrlen3mean) #ehrlen3condmn$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 provide 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)
#> NULL

The 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)
#> NULL

#### Step 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, including both the annual MPM and the mean. For the stochastic case, we will set the seed for R’s random number generation to make our output reproducible. Note that each $$\lambda$$ estimate includes a data frame describing the matrices in order (given as the labels object within the output list). Here is the set of ahistorical annual $$\lambda$$ estimates, followed by $$\lambda$$ for 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 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 ($$a = \text{log} \lambda _{S}$$) are 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 occasion 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 0.8863882
#> 2   1     1 0.9852582
writeLines("Deterministic mean historical: ")
#> Deterministic mean historical:
lambda3(ehrlen3mean)
#>   pop patch    lambda
#> 1   1     1 0.9180775
writeLines("Stochastic historical: ")
#> Stochastic historical:
set.seed(42)
slambda3(ehrlen3red)
#>   pop patch           a        var        sd          se
#> 1   1     1 -0.08893348 0.01748712 0.1322389 0.001322389

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
#> 6      1          3          3      Tm      Tm 0.2406242

Here we see that about 24% of the population is expected to be composed of tiny adults maintaining themselves as tiny adults, and that this stage pair is more numerous than any involving dormant seeds.

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 occasion 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 $ahist element of the historical stable stage distribution object. Notice below that the ss_prop column shows values that are a bit different from the ahistorical case, suggesting the influence of individual history, particularly showing that tiny adults are a larger share of the stage stage structure. ehrlen3mss$ahist
#>   matrix stage_id stage    ss_prop
#> 1      1        1    Sd 0.26017180
#> 2      1        2   Sdl 0.04072254
#> 3      1        3    Tm 0.29495112
#> 4      1        4    Sm 0.19905587
#> 5      1        5    La 0.06610243
#> 6      1        6   Flo 0.09222965
#> 7      1        7  Dorm 0.04676659

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$stage barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage", ylim = c(0, 0.35), col = c("black", "orangered", "grey", "darkred"), bty = "n") legend("topright", c("det ahistorical", "det historical", "stoc ahistorical", "stoc historical"), pch = 22, col = "black", pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n") Let’s take the deterministic portion first. Accounting for individual history increased the prevalence of tiny and small adults, but decreased the prevalence of dormant seeds, and dormant, large, and flowering adults. Now when we also take in the impact of temporal stochasticity, we can see differences in the proportion of dormant seeds, seedlings, and all adult stages, with the greatest differences in dormant seeds and tiny adults. Let’s take a look at the reproductive values now, in similar order to the stable stage distribution case. Initially, we will create all sets of reproductive value objects, and then we will plot them. The structure of these objects is the same as that of the stable stage structure outputs, and, in the case of the ahistorical MPM, the reproductive values are also the same as those produced by popbio’s reproductive.value() function. Because the four vectors are all standardized such that the first non-zero reproductive value is set to 1.0, they are on different scales, and so we will make them comparable for plotting purposes by standardizing them relative to their respective maxima. ehrlen2mrv <- repvalue3(ehrlen2mean) ehrlen3mrv <- repvalue3(ehrlen3mean) ehrlen2mrv_s <- repvalue3(ehrlen2, stochastic = TRUE, seed = 42) ehrlen3mrv_s <- repvalue3(ehrlen3red, stochastic = TRUE, seed = 42) rv_put_together <- cbind.data.frame((ehrlen2mrv$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("topleft", c("det ahistorical", "det historical", "stoc ahistorical",
"stoc historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")

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 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 = -2.5, x = seq(from = 0,
to = 1.19*length(ehrlen3mrv$hist$stage_2), by = 1.2), labels, xpd=TRUE, srt=45)

In the bar plot, the labels show the stages in the order of stage in occasion t followed by stage in occasion 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 ahistorical analysis, in which they were predicted to have the second highest reproductive value.

Now we will look at the 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("\nThis occurs in element ")
#>
#> This occurs in element
which(ehrlen2sens$ah_sensmats[[1]] == max(ehrlen2sens$ah_sensmats[[1]]))
#> [1] 6

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] 27 The 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. ehrlen3sens <- sensitivity3(ehrlen3mean) #ehrlen3sens 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.

Our historical matrices are large and full of 0s. So, we will look for the highest sensitivity associated with a biologically plausible element (i.e. non-zero matrix elements). Then, we will assess the highest biologically plausible sensitivity in the historically-corrected sensitivity matrices, to compare against the ahistorical sensitivity analysis.

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] 0.3139377
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$A[[1]] > 0)])) #> [1] 157 writeLines("\nThe highest historically-corrected sensitivity value among biologically plausible elements is: ") #> #> The highest historically-corrected sensitivity value among biologically plausible elements is: max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]) #> [1] 0.6219494 writeLines("\nThis occurs in element ") #> #> This occurs in element which(ehrlen3sens$ah_sensmats[[1]] == max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))
#> [1] 19

The maximum biologically plausible sensitivity value in the historical matrix appears to be element 157, which is associated with column 6 (tiny adult in occasions t-1 and t) and row 7 (tiny adult in occasion t to small adult in occasion t+1). This transition is from tiny adult in occasions t-1 and t to small adult in t+1. The historically-corrected sensitivity analysis finds that $$\lambda$$ is most sensitive to element 19, which is the transition from tiny adult to large adult. This is a little different from the ahistorical MPM, which suggested element 27 (small adult to flowering adult).

Before moving on, let’s repeat the above exercise with stochastic sensitivities. We will compare ahistorical vs. historical/historically-corrected analyses.

ehrlen2sens_s <- sensitivity3(ehrlen2, stochastic = TRUE)

writeLines("\nHighest stochastic sensitivity among biologically plausible elements (ahistorical): ")
#>
#> Highest stochastic sensitivity among biologically plausible elements (ahistorical):
max(ehrlen2sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])
#> [1] 0.4992827
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen2sens_s$ah_sensmats[[1]] == max(ehrlen2sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])) #> [1] 27 ehrlen3sens_s <- sensitivity3(ehrlen3red, stochastic = TRUE) writeLines("\nHighest stochastic sensitivity among biologically plausible elements (historical): ") #> #> Highest stochastic sensitivity among biologically plausible elements (historical): max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3red$A[[1]] > 0)]) #> [1] 0.4041625 writeLines("\nThis value is associated with element: ") #> #> This value is associated with element: which(ehrlen3sens_s$h_sensmats[[1]] == max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3red$A[[1]] > 0)]))
#> [1] 157

writeLines("\nHighest stochastic sensitivity among biologically plausible elements (historically-corrected): ")
#>
#> Highest stochastic sensitivity among biologically plausible elements (historically-corrected):
max(ehrlen3sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])
#> [1] 0.7965663
writeLines("\nThis value is associated with element: ")
#>
#> This value is associated with element:
which(ehrlen3sens_s$ah_sensmats[[1]] == max(ehrlen3sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])) #> [1] 19 Intriguingly, we find complete agreement in these baseline inferences between deterministic vs. stochastic analyses, and the major disagreement comes from differences between ahistorical vs. historical analyses. An alternative, or complementary, approach to perturbation 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 type, 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 (of course, the reality is that they answer different questions). Let’s 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$ah_elasmats[[1]]) #> [1] 0.167339 writeLines("\nThis value is associated with element: ") #> #> This value is associated with element: which(ehrlen2elas$ah_elasmats[[1]] == max(ehrlen2elas$ah_elasmats[[1]])) #> [1] 41 Elasticity 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 0.0586

Here we see that $$\lambda$$ is most strongly elastic in response to changes in transitions associated with flowering adults, followed by transitions involving small adults. Dormant seeds and seedlings have the smallest impact on $$\lambda$$, and the impacts of fecundity appear quite small.

Now on 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$h_elasmats[[1]]) #> [1] 0.1716148 writeLines("\nThis value is associated with element: ") #> #> This value is associated with element: which(ehrlen3elas$h_elasmats[[1]] == max(ehrlen3elas$h_elasmats[[1]])) #> [1] 156 The 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). 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$ah_elasmats, digits = 3) #> [[1]] #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] #> [1,] 0.00691 0.000000 0.00000 0.0000 0.00e+00 0.01148 0.00000 #> [2,] 0.01148 0.000000 0.00000 0.0000 0.00e+00 0.00000 0.00000 #> [3,] 0.00000 0.010613 0.19850 0.0344 8.31e-05 0.00000 0.00219 #> [4,] 0.00000 0.000288 0.04493 0.1617 2.56e-02 0.02788 0.01908 #> [5,] 0.00000 0.000000 0.00000 0.0306 4.54e-02 0.05420 0.00920 #> [6,] 0.00000 0.000000 0.00000 0.0367 6.08e-02 0.16439 0.00498 #> [7,] 0.00000 0.000576 0.00237 0.0162 7.43e-03 0.00891 0.00315 The historical matrices we used as input are reduced, but the historically-corrected output is in the original dimensions corresponding to the number of stages listed in the stageframe. This makes it easy to compare against the actual ahistorical elasticity matrix, for example by subtracting one matrix from the other to see the differences. We can also subtract the column sums in this exercise to assess overall shifts in the importance of stages. print((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]]), digits = 3) #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] #> [1,] 0.000361 0.000000 0.000000 0.00000 0.000000 -0.000146 0.00000 #> [2,] -0.000146 0.000000 0.000000 0.00000 0.000000 -0.020630 0.00000 #> [3,] 0.000000 -0.017223 0.046820 0.01803 -0.000437 0.000000 -0.00196 #> [4,] 0.000000 -0.000979 0.003827 0.00309 0.003531 0.011179 -0.00275 #> [5,] 0.000000 0.000000 -0.000604 0.00158 -0.003081 0.001022 -0.00824 #> [6,] 0.000000 0.000000 0.000000 0.00138 -0.007814 -0.002951 -0.00389 #> [7,] 0.000000 -0.002574 -0.004807 -0.00618 -0.001527 -0.001748 -0.00313 writeLines("\nColumn sums of differences:") #> #> Column sums of differences: colSums((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]])) #> [1] 0.0002153035 -0.0207764215 0.0452352007 0.0178973180 -0.0093278680 -0.0132751497 -0.0199683829 Accounting 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 create a barplot of the elasticities of life history stages from ahistorical vs. historically-corrected analyses. We will also incorporate stochastic elasticity analysis here to assess the importance of temporal environmental stochasticity on population growth. 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 ahistorical", "det historical", "stoc ahistorical", "stoc historical"), pch = 22, col = "black", pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n") We see the overall importance of both history and temporal environmental stochasticity here. Historical analyses generally find that population growth rate is less elastic in response to seedlings, and large and dormant adults, and more elastic to tiny and small adults than in ahistorical 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 transitions. 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 ahistorical", "det historical", "stoc ahistorical",
"stoc historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")

We see similar patterns across types of elasticity analysis. Particularly, population growth rate is most elastic in response to changes in stasis transitions, and least elastic to changes in fecundity.

Package lefko3 also includes functions to conduct deterministic and stochastic life table response experiments, and a general projection function that can be used for analyses such as quasi-extinction analysis. Users wishing to conduct these analyses should see our other vignettes, including the long-format vignettes on the projects page of the Shefferson lab website.

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.

## Acknowledgements

We are grateful to two anonymous reviewers whose scrutiny improved the quality of this vignette. The project resulting in this package and this tutorial was funded by Grant-In-Aid 19H03298 from the Japan Society for the Promotion of Science.

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