# BIOL 521: PVA by Stochastic Leslie Matrix Projection # Approach using matrix entries that specify a # mean and variance (and a specific distribution) # for each demographic variable in the matrix # Scott Creel 11/17/2011 # INTRODUCTION # This exercise builds a stochastic Leslie matrix to estimate extinction risk for # African wild dogs, using the data from Tanzania's Selous Game Reserve that we used # in class to build a life table and estimate lambda. # # In the previous exercise you saw the 'multiple matrices' approach to # stochastic projection... This approach assembles a set of Leslie matrices and randomly # draws one entire matrix for each step of projection. This approach assumes that all of the # demographic parameters covary -- they ALL change at once. # This assumption might not be correct, so you might want to do the projection in a manner that allows the # demographic parameters to vary independently of one another. To do this, # you must make random draws from a distribution for each demographic parameter, # rather than randomly drawing whole matrices. # # All that this requires is: # 1. Create a Leslie matrix in which each of the demographic parameters is specified by # a distribution rather than a constant. You do this by specifying a distribution, a mean and a variance. # There are other logical possibilities, but we'll use a binomial distribution for survival # rates and a normal distribution (truncated to be non-negative) for fecundities. # 2. Making a random draw from those distributions at each time step to get a 'stochastic Leslie matrix'. # 3. Projecting next year's population (tracked as a vector with the number # of individuals in each age class) with the stochastic Leslie matrix, just the same way you # would with a simple Leslie matrix with the mean for each demographic variable. # PART 1: BASIC DEMOGRAPHIC CALCULATIONS, SURVIVORSHIP CURVE. # These are the African wild dog data we used to build a life table in class. # Enter lx data (survivorship from birth) and examine survivorship curve lx <- c(1,0.75,0.63,0.49,0.37,0.29,0.13,0.09,0.02,0.02,0.02) log.lx <- log(lx) # create age variable x that corresponds to the lx values above x <- 0:10 plot(x,log.lx,main="Survivorship Curve",xlab="Age Class - x", ylab="Survivorship - log(lx)",pch=16) # add a line for perfect type II survivorship, just for visual comparison # type II survivorship is the case with a constant survival rate (sx) # for all age classes, which yields a straight line # on a survivorship curve with a log-Y axis lines(x,(-3.9/10*x),type="l",lty=2) # Enter mx data (fecundity) and calculate approximate growth rate # These values differ from the ones in the life table in class, because they # are for a post-birth pulse annual census approach mx <- c(0,0,0.1,0.28,0.59,1.12,3.99,2.14,3.32,2.66,0) contrib.growth <- lx*mx Ro <- sum(contrib.growth) x.lxmx <- x*contrib.growth T <- sum(x.lxmx)/Ro r.app <- log(Ro)/T Ro T r.app # Use euler equation to check if the approximate solution for r is reasonable. # If the sum is close to one, the approximate solution is fairly accurate. # You Can get exact solution by altering the estimated value of r until the sum # equals one, but can more easily get exact lambda [see below] as the dominant # Eigenvalue of the Leslie matrix sum(exp(-r.app*x)*lx*mx) # PART 2: DETERMINISTIC LESLIE PROJECTION # convert lx data to sx values Sx <- c(73/97,61/73,48/61,36/48,28/36,13/28,9/13,2/9,2/2,2/2,0/2) #convert mx to Fx (here for a post-birth pulse count) Fx <- mx*Sx[1] # fecundity values S <- Sx[-1] # dropping the first value F <- Fx[-1] # create an empty matrix (with all zeros) of the correct dimensions. les.mat <- matrix(rep(0,10*10),nrow=10) # Fill in relevant values for the non-zero entries # -- row 1 is fecundities les.mat[1,] <- F # -- and the subdiagonal is annual survival rates, using a loop to assign them to # the correct positions within the matrix for(i in 1:9){ les.mat[(i+1),i] <- S[i] } #examine the completed projection matrix: Fx values in the top row, Sx in subdiagonal les.mat # Deterministic projection with Leslie matrix. This script does this in two ways # and compares results as a check. First you'll do it for yourself, and then use # the popbio package, which has functions that make it much easier # First, load the popbio package. library(popbio) # Input a vector with the initial distribution of individuals # in across age classes. In this example, 97 newborns only. N <- c(97,0,0,0,0,0,0,0,0,0) # Create an empty matrix [with NA values] to store the set of projected population vectors # for 30 years of projection into the future, starting with population # defined by N above. Dist_Year <- matrix(NA,nrow=31,ncol=10) # Set the initial population in the first year. Dist_Year[1,] <- N # Do the matrix multiplication for 30 years of projection, years 2 to 31 # matrix multiplication uses %*% rather than * for(i in 2:31) { Dist_Year[i,] <- les.mat %*% Dist_Year[i-1,] } # Calculate and examine total size at each time step. In the apply() function, the sum argument # species to calculate a sum, the 1 specifes that rows should be summed (2 would specify to sum # columns), and the Dist_Year indicates what object to do all this on. This is a very old-school # line of code, and there are other ways to do this. proj <- apply(Dist_Year,1,sum) proj # Double-check using the Popbio package function 'pop.projection'. p <- pop.projection(les.mat,N,31) p # Examine the results. Note that results for popualtion size agree exactly, and that the # pop.projection function also provides a lot of other useful output, # including lambda (dominant eigenvalue of the matrix) and the stable # age distribution (R eigenvector of the matrix). The stable age distribution # output is output as the proportion of the population in each age class. # Popbio includes a function eigen.analysis that will calculate the elasticities. # Explore. Remember that you can press F1 with the cursor in a function to get help on it. # PART 3: STOCHASTIC LESLIE PROJECTION # This uses 4 nested loops. The i and j loops are innermost and draw stochastic fecundity and survival. # The t loop is next, and steps across 30 years of projection from an initial N vector # the a loop is outermost and replicates the 30 years of projection for 100 iterations. tot <- rep(NA,100) # Dummy vector to store total population size for 100 replicates t <- 31 # Number of years for the projection. for(a in 1:100){ # Repeat stochastic 't'-year projections 100 times. stoc_year <- matrix(NA,nrow=t,ncol=10) # Dummy matrix of age-class values. One row per time step. Columns are # individuals in each age class. N <- c(97,rep(0,9)) # Initial values in each age-class. stoc_year[1,] <- N # Assign these values to first row in age-class matrix. for(i in 2:t){ # Begin loop for projections. lm <- les.mat # Set up a temporary Leslie matrix for each iteration # with the correct mean fecundities and survival rates for(j in 1:10){ # Randomly draw fecundities for each class. lm[1,j] <- rpois(1,les.mat[1,j]) # Mean of Poisson is fecundity value. } # Ends the 'j' loop that draws fecundities. for(k in 1:9){ # Randomly draw survival probabilities for each class. n <- stoc_year[(i-1),k] # n is number of individuals to calculate live/die for. lm[(k+1),k] <- ifelse(n>1,rbinom(1,size=round(n),p=les.mat[(k+1),k])/n,0) # need ifelse statement to deal with the possibility # that there are no individuals in that age class. } # Ends the 'k' loop to draw survival rates. stoc_year[i,] <- lm %*% stoc_year[(i-1),] # Matrix multiplication for next time-step. } # Ends the 'i' loop projecting across time. tot[a] <- sum(stoc_year[t,]) } # Ends the 'a' loop of one hundred iterations of the projections ci <- quantile(tot,probs=c(0.025,0.975)) # Find 95% confidence intervals on final population size. # Examine final population size for each of 100 iterations, and the 95% confidence interval on final population size # This is stochastic so I can't tell you exactly what numbers you'll get, but your confidence interval # on population size 30 years from now probably does not overlap zero. tot ci