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LTREs, sLTREs, and MPM import withAnthyllis vulneraria

Richard P. Shefferson and Raziel Davison

In this vignette, we will import a MPM created elsewhere intolefko3 and analyze it via LTRE and sLTRE analysis. Theseare inherently ahistorical MPMs, and so we do not include historicalanalyses in this vignette.

To reduce vignette size, we have prevented some statements from runningif they produce long stretches of output. Examples include mostsummary() calls. In these cases, we include hashtaggedversions of these calls, and we encourage the user to run thesestatements without hashtags to examine the output.

This vignette is only a sample analysis. Detailed information andinstructions on usinglefko3 are available through a freeonline e-book calledlefko3: agentle introduction, as well as through the resources found onthe{r}evolutionarydemography website.

ORGANISM AND POPULATIONS

Davison et al. (2010) reported stochasticcontributions made by differences in vital rate means and variancesamong nine natural populations of the perennial herbAnthyllisvulneraria, also known as kidney vetch, occurring in calcareousgrasslands in the Viroin Valley of southwestern Belgium.A.vulneraria is a grassland specialist and the unique host plant ofthe Red-listed blue butterfly (Cupido minimus). It is ashort-lived, rosette-forming legume with a complex life cycle includingstasis and retrogression between four stages but no seedbank (seedlings,juveniles, small adults and large adults; Figure 9.1).

Figure 9.1. Lifehistory model ofAnthyllis vulneraria. Solid arrows indicatesurvival transitions while dashed arrows indicate fecunditytransitions.

Nine populations (N = 27-50,000) growing in distinct grassland fragmentswere surveyed between 2003 and 2006, yielding three (4x4) annualtransition matrices for each population. The populations occurred withingrassland fragments, and were mostly managed as nature reserves throughrotational sheep grazing. These surveys coincided with a summer heatwave (2003), followed by a spring drought (2005) and an even moreextreme heat wave (2006). These populations have been subject todetailed study for aspects of their genetics and demography, and furtherdetails on the sites can be obtained through the resulting publications(Krauss, Steffan-Dewenter, and Tscharntke 2004;Honnay et al. 2006; Piessens et al. 2009).

BASIC WORKFLOW

Our goal in this exercise will be to import the published MPMs availablefor these nine populations ofAnthyllis vulneraria, and toanalyze the demographic differences between populations usingdeterministic and stochastic life table response experiments (LTRE andsLTRE). We will use the matrices analyzed inDavison et al. (2010), and attempt to reproducetheir results.

Step 1. Life history model development

We will first describe the life history characterizing the dataset withastageframe. Since we do not have the original demographicdataset that produced the published matrices, we do not need to know theexact sizes of plants and so will use proxy values. These proxy valuesneed to be unique and non-negative, and they need non-overlapping binsusable as size classes defining each stage. However, since we are notanalyzing size itself, they do not need any further basis in reality.Other characteristics must be exact and realistic to make sure that theanalyses work properly, including all other stage descriptions such asreproductive status, propagule status, and observation status. Note thatwe are using the midpoint approach to determining the size bins here,using the default bin halfwidths of 0.5. However, we could have used thesizemin andsizemax options to moredeliberately set the size bin minima and maxima instead.

sizevector<-c(1,1,2,3)# These sizes are not from the original paperstagevector<-c("Sdl","Veg","SmFlo","LFlo")repvector<-c(0,0,1,1)obsvector<-c(1,1,1,1)matvector<-c(0,1,1,1)immvector<-c(1,0,0,0)propvector<-c(0,0,0,0)indataset<-c(1,1,1,1)binvec<-c(0.5,0.5,0.5,0.5)comments<-c("Seedling","Vegetative adult","Small flowering","Large flowering")anthframe<-sf_create(sizes = sizevector,stagenames = stagevector,repstatus = repvector,obsstatus = obsvector,matstatus = matvector,immstatus = immvector,indataset = indataset,binhalfwidth = binvec,propstatus = propvector,comments = comments)#anthframe

Step 2. Data organization

Next we will enter the data for this vignette. Our data is in the formof matrices published inDavison et al.(2010). We will enter these matrices as standard Rmatrix class objects. All matrices are square with 4columns, and we fill them by row. As an example, let’s load the firstmatrix.

XC3<-matrix(c(0,0,1.74,1.74,# POPN C 2003-20040.208333333,0,0,0.057142857,0.041666667,0.076923077,0,0,0.083333333,0.076923077,0.066666667,0.028571429),4,4,byrow =TRUE)XC3#>            [,1]       [,2]       [,3]       [,4]#> [1,] 0.00000000 0.00000000 1.74000000 1.74000000#> [2,] 0.20833333 0.00000000 0.00000000 0.05714286#> [3,] 0.04166667 0.07692308 0.00000000 0.00000000#> [4,] 0.08333333 0.07692308 0.06666667 0.02857143

These areA matrices, meaning that they include allsurvival-transitions and fecundity for the population as a whole. ThecorrespondingU andF matrices were notprovided in that paper, although it is most likely that the elementsvalued at1.74 in the top-right corner are fecundityelements while the rest of the matrix is composed only of survivaltransitions. The order of rows and columns corresponds to the order ofstages in the stageframeanthframe. Let’s now input theremaining matrices.

XC4<-matrix(c(0,0,0.3,0.6,# POPN C 2004-20050.32183908,0.142857143,0,0,0.16091954,0.285714286,0,0,0.252873563,0.285714286,0.5,0.6),4,4,byrow =TRUE)XC5<-matrix(c(0,0,0.50625,0.675,# POPN C 2005-20060,0,0,0.035714286,0.1,0.068965517,0.0625,0.107142857,0.3,0.137931034,0,0.071428571),4,4,byrow =TRUE)XE3<-matrix(c(0,0,2.44,6.569230769,# POPN E 2003-20040.196428571,0,0,0,0.125,0.5,0,0,0.160714286,0.5,0.133333333,0.076923077),4,4,byrow =TRUE)XE4<-matrix(c(0,0,0.45,0.646153846,# POPN E 2004-20050.06557377,0.090909091,0.125,0,0.032786885,0,0.125,0.076923077,0.049180328,0,0.125,0.230769231),4,4,byrow =TRUE)XE5<-matrix(c(0,0,2.85,3.99,# POPN E 2005-20060.083333333,0,0,0,0,0,0,0,0.416666667,0.1,0,0.1),4,4,byrow =TRUE)XF3<-matrix(c(0,0,1.815,7.058333333,# POPN F 2003-20040.075949367,0,0.05,0.083333333,0.139240506,0,0,0.25,0.075949367,0,0,0.083333333),4,4,byrow =TRUE)XF4<-matrix(c(0,0,1.233333333,7.4,# POPN F 2004-20050.223880597,0,0.111111111,0.142857143,0.134328358,0.272727273,0.166666667,0.142857143,0.119402985,0.363636364,0.055555556,0.142857143),4,4,byrow =TRUE)XF5<-matrix(c(0,0,1.06,3.372727273,# POPN F 2005-20060.073170732,0.025,0.033333333,0,0.036585366,0.15,0.1,0.136363636,0.06097561,0.225,0.166666667,0.272727273),4,4,byrow =TRUE)XG3<-matrix(c(0,0,0.245454545,2.1,# POPN G 2003-20040,0,0.045454545,0,0.125,0,0.090909091,0,0.125,0,0.090909091,0.333333333),4,4,byrow =TRUE)XG4<-matrix(c(0,0,1.1,1.54,# POPN G 2004-20050.111111111,0,0,0,0,0,0,0,0.111111111,0,0,0),4,4,byrow =TRUE)XG5<-matrix(c(0,0,0,1.5,# POPN G 2005-20060,0,0,0,0.090909091,0,0,0,0.545454545,0.5,0,0.5),4,4,byrow =TRUE)XL3<-matrix(c(0,0,1.785365854,1.856521739,# POPN L 2003-20040.128571429,0,0,0.010869565,0.028571429,0,0,0,0.014285714,0,0,0.02173913),4,4,byrow =TRUE)XL4<-matrix(c(0,0,14.25,16.625,# POPN L 2004-20050.131443299,0.057142857,0,0.25,0.144329897,0,0,0,0.092783505,0.2,0,0.25),4,4,byrow =TRUE)XL5<-matrix(c(0,0,0.594642857,1.765909091,# POPN L 2005-20060,0,0.017857143,0,0.021052632,0.018518519,0.035714286,0.045454545,0.021052632,0.018518519,0.035714286,0.068181818),4,4,byrow =TRUE)XO3<-matrix(c(0,0,11.5,2.775862069,# POPN O 2003-20040.6,0.285714286,0.333333333,0.24137931,0.04,0.142857143,0,0,0.16,0.285714286,0,0.172413793),4,4,byrow =TRUE)XO4<-matrix(c(0,0,3.78,1.225,# POPN O 2004-20050.28358209,0.171052632,0,0.166666667,0.084577114,0.026315789,0,0.055555556,0.139303483,0.447368421,0,0.305555556),4,4,byrow =TRUE)XO5<-matrix(c(0,0,1.542857143,1.035616438,# POPN O 2005-20060.126984127,0.105263158,0.047619048,0.054794521,0.095238095,0.157894737,0.19047619,0.082191781,0.111111111,0.223684211,0,0.356164384),4,4,byrow =TRUE)XQ3<-matrix(c(0,0,0.15,0.175,# POPN Q 2003-20040,0,0,0,0,0,0,0,1,0,0,0),4,4,byrow =TRUE)XQ4<-matrix(c(0,0,0,0.25,# POPN Q 2004-20050,0,0,0,0,0,0,0,1,0.666666667,0,1),4,4,byrow =TRUE)XQ5<-matrix(c(0,0,0,1.428571429,# POPN Q 2005-20060,0,0,0.142857143,0.25,0,0,0,0.25,0,0,0.571428571),4,4,byrow =TRUE)XR3<-matrix(c(0,0,0.7,0.6125,# POPN R 2003-20040.25,0,0,0.125,0,0,0,0,0.25,0.166666667,0,0.25),4,4,byrow =TRUE)XR4<-matrix(c(0,0,0,0.6,# POPN R 2004-20050.285714286,0,0,0,0.285714286,0.333333333,0,0,0.285714286,0.333333333,0,1),4,4,byrow =TRUE)XR5<-matrix(c(0,0,0.7,0.6125,# POPN R 2005-20060,0,0,0,0,0,0,0,0.333333333,0,0.333333333,0.625),4,4,byrow =TRUE)XS3<-matrix(c(0,0,2.1,0.816666667,# POPN S 2003-20040.166666667,0,0,0,0,0,0,0,0,0,0,0.166666667),4,4,byrow =TRUE)XS4<-matrix(c(0,0,0,7,# POPN S 2004-20050.333333333,0.5,0,0,0,0,0,0,0.333333333,0,0,1),4,4,byrow =TRUE)XS5<-matrix(c(0,0,0,1.4,# POPN S 2005-20060,0,0,0,0,0,0,0.2,0.111111111,0.75,0,0.2),4,4,byrow =TRUE)

We will incorporate all of these matrices into alefkoMatobject. To do so, we will first organize the matrices in the properorder within a list. Then, we will create thelefkoMatobject to hold these matrices by calling functioncreate_lM() with the list object we created. We will alsoinclude metadata describing the order of populations (here treated aspatches), and the order of monitoring occasions.

mats_list<-list(XC3, XC4, XC5, XE3, XE4, XE5, XF3, XF4, XF5, XG3, XG4, XG5,  XL3, XL4, XL5, XO3, XO4, XO5, XQ3, XQ4, XQ5, XR3, XR4, XR5, XS3, XS4, XS5)anth_lefkoMat<-create_lM(mats_list, anthframe,hstages =NA,historical =FALSE,poporder =1,patchorder =c(1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9),yearorder =c(2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005,2003,2004,2005))#> Warning: No supplement provided. Assuming fecundity yields all propagule and immature stages.#anth_lefkoMat

The resulting object has all of the elements of a standardlefkoMat object except for those elements related toquality control in the demographic dataset and linear modeling. TheoptionUFdecomp was left at its default(UFdecomp = TRUE), socreate_lM() used thestageframe to infer where fecundity values should be located and createdU andF matrices accordingly. The defaultoption forhistorical is set toFALSE,yielding anNA in place of thehstageselement, which would typically list the order of historical stage pairs.

#summary(anth_lefkoMat)

The summary of this newlefkoMat object shows us that wehave 27 matrices with 4 rows and columns each. The estimated numbers oftransitions corresponds to the non-zero entries in each matrix. We seethat we are covering 1 population, 9 patches, and 3 time steps. All ofthe survival probabilities observed fell within the bounds of 0 to 1, soeverything seems alright.

Step 3. MPM analysis

Let’s develop arithmetic mean matrices and assess the deterministic andstochastic population growth rates,\(\lambda\) and\(a\).

anth_lmean<-lmean(anth_lefkoMat)lambda2<-lambda3(anth_lefkoMat)lambda2m<-lambda3(anth_lmean)set.seed(42)sl2<-slambda3(anth_lefkoMat)#Stochastic growth ratesl2$expa<-exp(sl2$a)plot(lambda~ year2,data =subset(lambda2, patch==1),ylim =c(0,2.5),xlab ="Year",ylab =expression(lambda),type ="l",col ="gray",lty=2,lwd =2,bty ="n")lines(lambda~ year2,data =subset(lambda2, patch==2),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==3),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==4),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==5),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==6),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==7),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==8),col ="gray",lty=2,lwd =2)lines(lambda~ year2,data =subset(lambda2, patch==9),col ="gray",lty=2,lwd =2)abline(a = lambda2m$lambda[1],b =0,lty =1,lwd =4,col ="orangered")abline(a = sl2$expa[1],b =0,lty =1,lwd =4,col ="darkred")legend("topleft",c("det annual","det mean","stochastic"),lty =c(2,1,1),col =c("gray","orangered","darkred"),lwd =c(2,4,4),bty ="n")
Figure 9.2. Deterministic vs. stochastic lambda
Figure 9.2. Deterministic vs. stochasticlambda

These populations exhibit highly variable growth across the short fieldstudy during which they were monitored. They also appear to bedeclining, as shown by\(\lambda\) forthe overall mean matrix, and\(e\) tothe power of the stochastic growth rate\(a =\text{log} \lambda\).

Let’s conduct a life table response experiment (LTRE) next. This will bea deterministic LTRE, meaning that we will assess the impacts ofdifferences in matrix elements between the core matrices input via thelefkoMat objectanth_lefkoMat and the overallarithmetic grand mean matrix on differences in the deterministic growthrate,\(\lambda\). We could use adifferent reference matrix if we wished, but the default is to use thearithmetic grand mean. Here, because we cannot analyze temporal shiftsusing a deterministic LTRE, we will utilize the mean matrix set instead.

trialltre_det<-ltre3(anth_lmean,sparse ="auto")#> Warning: Matrices input as mats will also be used in reference matrix calculation.#> Using all refmats matrices in reference matrix calculation.#trialltre_det

The resultinglefkoLTRE object gives the LTRE contributionsfor each matrix relative to the arithmetic grand mean matrix. While thedifferences in LTRE contributions across space are of interest, wecannot infer differences across time this way because matrices are notrelated additively across time. To assess the contributions across spacewhile accounting for temporal shifts, we should conduct a stochasticLTRE (sLTRE) or small-noise approximation LTRE (SNA-LTRE) on theoriginal annual matrices. Here, conduct a sLTRE.

trialltre_sto<-ltre3(anth_lefkoMat,stochastic =TRUE,times =10000,tweights =NA,sparse ="auto",seed =42)#> Warning: Matrices input as mats will also be used in reference matrix calculation.#> Using all refmats matrices in reference matrix calculation.#trialltre_sto

The sLTRE produces output that is a bit different from the deterministicLTRE output. In the output, we see two lists of matrices prior to theMPM metadata. The first,cont_mean, is a list of matricesshowing the contributions of differences in mean elements between thepatch-level temporal mean matrices and the reference temporal meanmatrix. The second,cont_sd, is a list of matrices showingthe contributions of differences in the temporal standard deviation ofeach element between the patch-level and reference matrix sets. In otherwords, while a standard LTRE shows the impact of changes in matrixelements on\(\lambda\), the sLTREshows the impacts of changes in the temporal mean and variability ofmatrix elements on\(\text{log}\lambda\). Thelabels element shows the order ofmatrices with reference to the populations or patches (remember thathere, the populations are referred to as patches).

The output objects are large and can take a great deal of effort to lookover and understand. Therefore, we will show three approaches toassessing these objects, using an approach similar to that used toassess elasticities. These methods can be used to assess patterns in all9 populations, but for brevity we will focus only on the firstpopulation here. First, we will identify the elements most stronglyimpacting the population growth rate in each case.

# Highest (i.e most positive) deterministic LTRE contribution:max(trialltre_det$cont_mean[[1]])#> [1] 0.02099388# Highest deterministic LTRE contribution is associated with element:which(trialltre_det$cont_mean[[1]]==max(trialltre_det$cont_mean[[1]]))#> [1] 3# Lowest (i.e. most negative) deterministic LTRE contribution:min(trialltre_det$cont_mean[[1]])#> [1] -0.2786045# Lowest deterministic LTRE contribution is associated with element:which(trialltre_det$cont_mean[[1]]==min(trialltre_det$cont_mean[[1]]))#> [1] 13# Highest stochastic mean LTRE contribution:max(trialltre_sto$cont_mean[[1]])#> [1] 0.01788772# Highest stochastic mean LTRE contribution is associated with element:which(trialltre_sto$cont_mean[[1]]==max(trialltre_sto$cont_mean[[1]]))#> [1] 3# Lowest stochastic mean LTRE contribution:min(trialltre_sto$cont_mean[[1]])#> [1] -0.3700463# Lowest stochastic mean LTRE contribution is associated with element:which(trialltre_sto$cont_mean[[1]]==min(trialltre_sto$cont_mean[[1]]))#> [1] 13# Highest stochastic SD LTRE contribution:max(trialltre_sto$cont_sd[[1]])#> [1] 0.00952507# Highest stochastic SD LTRE contribution is associated with element:which(trialltre_sto$cont_sd[[1]]==max(trialltre_sto$cont_sd[[1]]))#> [1] 13# Lowest stochastic SD LTRE contribution:min(trialltre_sto$cont_sd[[1]])#> [1] -0.01135741# Lowest stochastic SD LTRE contribution is associated with element:which(trialltre_sto$cont_sd[[1]]==min(trialltre_sto$cont_sd[[1]]))#> [1] 16# Total positive deterministic LTRE contributions:sum(trialltre_det$cont_mean[[1]][which(trialltre_det$cont_mean[[1]]>0)])#> [1] 0.06338566# Total negative deterministic LTRE contributions:sum(trialltre_det$cont_mean[[1]][which(trialltre_det$cont_mean[[1]]<0)])#> [1] -0.4079573# Total positive stochastic mean LTRE contributions:sum(trialltre_sto$cont_mean[[1]][which(trialltre_sto$cont_mean[[1]]>0)])#> [1] 0.04459663# Total negative stochastic mean LTRE contributions:sum(trialltre_sto$cont_mean[[1]][which(trialltre_sto$cont_mean[[1]]<0)])#> [1] -0.5085542# Total positive stochastic SD LTRE contributions:sum(trialltre_sto$cont_sd[[1]][which(trialltre_sto$cont_sd[[1]]>0)])#> [1] 0.01026457# Total negative stochastic SD LTRE contributions:sum(trialltre_sto$cont_sd[[1]][which(trialltre_sto$cont_sd[[1]]<0)])#> [1] -0.02232758

The output for the deterministic LTRE shows that element 13, which isthe fecundity transition from large flowering adult to seedling (column4, row 1), has the strongest influence. This contribution is negative,so it reduces\(\lambda\). The mostpositive contribution to\(\lambda\) isfrom element 3, which is the growth transition from seedling to smallflowering adult (column 1, row 3). We also see the sLTRE produced thesame results in contributions of shifts in the mean as the deterministicLTRE. Variability in elements also contributes to shifts in\(\text{log} \lambda\), though less so thanshifts in mean elements. The strongest positive contribution is fromvariation in the fecundity transition from large flowering adult toseedling (element 13, column 4, row 1), while the most negativecontribution is from stasis as a large flowering adult (element 16, rowand column 4). A comparison of summed LTRE elements shows that negativecontributions of mean elements were most influential in the stochasticcase, while variability of elements had little overall influence.

Second, we will identify which stages exerted the strongest impact onthe population growth rate.

ltre_pos<- trialltre_det$cont_mean[[1]]ltre_neg<- trialltre_det$cont_mean[[1]]ltre_pos[which(ltre_pos<0)]<-0ltre_neg[which(ltre_neg>0)]<-0sltre_meanpos<- trialltre_sto$cont_mean[[1]]sltre_meanneg<- trialltre_sto$cont_mean[[1]]sltre_meanpos[which(sltre_meanpos<0)]<-0sltre_meanneg[which(sltre_meanneg>0)]<-0sltre_sdpos<- trialltre_sto$cont_sd[[1]]sltre_sdneg<- trialltre_sto$cont_sd[[1]]sltre_sdpos[which(sltre_sdpos<0)]<-0sltre_sdneg[which(sltre_sdneg>0)]<-0ltresums_pos<-cbind(colSums(ltre_pos),colSums(sltre_meanpos),colSums(sltre_sdpos))ltresums_neg<-cbind(colSums(ltre_neg),colSums(sltre_meanneg),colSums(sltre_sdneg))ltre_as_names<- trialltre_det$ahstages$stagebarplot(t(ltresums_pos),beside = T,col =c("black","grey","red"),ylim =c(-0.50,0.10))barplot(t(ltresums_neg),beside = T,col =c("black","grey","red"),add =TRUE)abline(0,0,lty=3)text(cex=1,y =-0.57,x =seq(from =2,to =3.98*length(ltre_as_names),by =4), ltre_as_names,xpd=TRUE,srt=45)legend("bottomleft",c("deterministic","stochastic mean","stochastic SD"),col =c("black","grey","red"),pch =15,bty ="n")
Figure 9.3. LTRE contributions by stage
Figure 9.3. LTRE contributions by stage

The plot shows that large flowering adults exerted the strongestinfluence on both\(\lambda\) and\(\text{log} \lambda\), with the latterinfluence being through the impact of shifts in the mean. This impact isoverwhelmingly negative. The next largest impact comes from smallflowering adults, in both cases the influence being negative on thewhole.

Finally, we will assess what transition types exert the greatest impacton population growth rate.

det_ltre_summary<-summary(trialltre_det)sto_ltre_summary<-summary(trialltre_sto)ltresums_tpos<-cbind(det_ltre_summary$ahist_mean$matrix1_pos,  sto_ltre_summary$ahist_mean$matrix1_pos,  sto_ltre_summary$ahist_sd$matrix1_pos)ltresums_tneg<-cbind(det_ltre_summary$ahist_mean$matrix1_neg,  sto_ltre_summary$ahist_mean$matrix1_neg,  sto_ltre_summary$ahist_sd$matrix1_neg)barplot(t(ltresums_tpos),beside = T,col =c("black","grey","red"),ylim =c(-0.55,0.10))barplot(t(ltresums_tneg),beside = T,col =c("black","grey","red"),add =TRUE)abline(0,0,lty =3)text(cex=0.85,y =-0.64,x =seq(from =2,to =3.98*length(det_ltre_summary$ahist_mean$category),by =4),    det_ltre_summary$ahist_mean$category,xpd=TRUE,srt=45)legend("bottomleft",c("deterministic","stochastic mean","stochastic SD"),col =c("black","grey","red"),pch =15,bty ="n")
Figure 9.4. LTRE contributions by transition type
Figure 9.4. LTRE contributions by transitiontype

The overall greatest impact on the population growth rate is fromfecundity, and these contributions are overwhelmingly negative. Clearlytemporal variation has strong effects here that deserve to be assessedproperly.

LTREs and sLTREs are powerful tools, and more complex versions of bothanalyses exist. Please consultCaswell(2001) andDavison et al. (2013)for more information. Also seeDavison et al.(2013) for information on the small noise approximation LTRE,which is also possible inlefko3.

Packagelefko3 also includes functions to conduct manyother analyses, including deterministic and stochastic life tableresponse experiments, and general projection including quasi-extinctionanalysis and density dependent analysis. Users wishing to conduct theseanalyses should see our free e-manual calledlefko3: agentle introduction, as well as through the resources found onthe{r}evolutionarydemography website.

Literature cited

Caswell, Hal. 2001.Matrix Population Models: Construction,Analysis, and Interpretation. Second edition. Sunderland,Massachusetts, USA: Sinauer Associates, Inc.
Davison, Raziel, Hans Jacquemyn, Dries Adriaens, Olivier Honnay, Hans deKroon, and Shripad Tuljapurkar. 2010.“Demographic Effects ofExtreme Weather Events on a Short-Lived Calcareous Grassland Species:Stochastic Life Table Response Experiments.”Journal ofEcology 98 (2): 255–67.https://doi.org/10.1111/j.1365-2745.2009.01611.x.
Davison, Raziel, Florence Nicole, Hans Jacquemyn, and ShripadTuljapurkar. 2013.“Contributions of Covariance: Decomposing theComponents of Stochastic Population Growth inCypripedium Calceolus.”The AmericanNaturalist 181 (3): 410–20.https://doi.org/10.1086/669155.
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