# BIOL 521: Count Based PVA
# Scott Creel 9/30/2011
#
# INTRODUCTION
# This lab implements the same count-based PVA approach that we examined in class using POPTOOLS
# in Excel. The data set comes from Eberhardt et al (1986) and Haroldson et al 1999, with
# monitoring data for Grizzly bears in Yellowstone National Park. Grizzly bears (Ursus arctos horribilus) are
# listed as threatened by the USFWS for most of the lower 48 states, with a few exceptions in areas where
# they are listed as 'experimental, non-essential'. (This status usually is associated with reintroduced
# carnivore populations, so that they can be killed in the case of livestock conflicts.)
# There is considerable discussion of delisting Grizzly bears in the Yellowstone region because of recent population
# growth, geographic expansion and demographic trends. Thus, this is an excellent example of a situation in which a
# PVA is useful.
#
# The basic data set is annual counts of 'unreplicated females with cubs'. This is an estimate of population size that
# includes information from several types of monitoring, including aerial counts and data from individually-known,
# radiocollared bears.
# CODE
# enter the years from which the annual counts come.
# because there are no missing years (nice!) we can use the seq function
year=seq(1959,1996)
########## OR
year=seq(from=1959,to=1996,by=1)
##########
# enter the counts of unreplicated females with cubs
N=c(44,47,46,44,46,45,46,40,39,39,42,39,41,40,33,36,34,39,35,34,38,36,37,41,39,51,
47,57,48,60,65,74,69,65,57,70,81,99)
#check that the data are entered correctly
a= length(year)
a
year
b=length(N)
b
N
# calculate annual growth rates
# recall that the intrinsic rate of increase (r)= ln(lambda), lambda = e^r
# and of course we can't get lambda for the lastyear
ln.lambda=log(N[-1]/N[-length(N)])
r=ln.lambda
# examine the growth rates
r
##############Summary of the data and inspection
summary(r)
boxplot(r)
hist(r)
################
# calculate mean & sd of log(lambda) values
# we need the geometric mean of the growth rates, as explained in class
geomean=mean(r)
#examine the geometric mean intrinsic rate of increase
geomean
# and the variances are calculated from the ln(lambda) values, because the growth rates
# are lognormally distributed
var.r=var(r)
sd.r=sqrt(var.r)
##############OR
sd.r=sd(r)
################
#examine the variance and SD of the intrinsic rate of increase
var.r
sd.r
# examine the actual population dynamics that were observed
plot(year,N, xlab = "Year", ylab = "Population size", pch = 19, ylim=c(0,100))
############## OR
plot(N~year, xlab = "Year", ylab = "Population size", pch = 19, ylim=c(0,100),
type="b", main="Grizzly Population size over years",cex.lab=1.5,
cex=1,cex.axis=1.5)
##############
# add an trajectory for simple exponental growth over the same period, with
# the observed geomean for r. Recall that the number of years of growth is one
# less than the number of annual counts. For each year (x), we need to determine
# Y = the expected population size with exponental growth = Nt = N0 * e^rt
lines(year,N[1]*exp(geomean)^(0:(length(N)-1)),lty=1,col='red')
############# OR
lines(year,N[1]*exp(geomean)^(0:(length(N)-1)),pch=19,col='red',type="b",cex=0.8)
legend(1960,100,c("Empirical data"," Exponential Model prediction"),lty=c(1,1),col=c("black","red"),pch=c(19,19), bty="y")
############
################
# The growth model matches the final population size. The model is accurate in that sense.
# It is also clear from the relationship of the actual points to the exponential growth
# model (red line) that the true dynamics were more complex than exponential growth
# with a constant per capita growth rate. The growth rate varied through time... broadly,it was
# negative for a span of years, then more strongly positive than the mean.
# Projections for a simple PVA should, at a minimum, take the observed variation in growth rates into account.
# Recall that the observed variation in growth rate, when calculated from Nt+1/Nt as we
# did, includes "sampling error" (apparent variation in growth that is not real). The real variation in the
# growth rate ("process error" or "process variance") is smaller than the sd.r value that we've
# calculated. A more thorough analysis (see Morris & Doak 2002, CH. 5) would partition out the process
# variance (exclduing sampling error) and use it for projections.
# For now, CONSIDER THE IMPLICATIONS FOR THE PROJECTED DYNAMICS of not removing the sampling error.
# project population dynamics across the observed span of years (hindcasting) and then 30 years into
# the future (forecasting)
N0 = N[1]
# Define a threshold population size that we consider unacceptable. For true local extirpation, this
# would be zero. I'm setting it equal to the population size in 1959.
threshold=44
# Set the number of years to project forward (checking match to 38 years of data, plus 30 years projection)
project=68
# Set the of model iterations to perform
runs=1000
# Create an empty matrix to hold the output (population size with stochastic exponential growth),
# for the specified number of years and iterations
stoch.pop=matrix(NA,project,runs)
stoch.pop[1,]=N0
# loop to create stochastic population sizes
for (i in 1:runs){ # looping over 1000 runs of the stochastic model
for (t in 2:project){ # and looping of 68 years of projection
lambda=exp(rnorm(1,geomean,sd.r)) # draw a value of lambda from a lognormal distribution
stoch.pop[t,i]=stoch.pop[(t-1),i]*lambda # and project one time step from the current pop size
if(stoch.pop[t,i]<=threshold) break # leave the loop if pop <= threshold
}
}
# examine the stochastic output (note that this is huge, 68 x 1000, so we'll just look at the first and
# last few rows of output)
head(stoch.pop)
tail(stoch.pop)
############### PERCENTAGE OF RUNS UNDER THRESHOLD POPULATION SIZE
percentage.under<-(runs-length(which(stoch.pop[project,]>=0)))/runs*100
percentage.under
## The WHICH command nested within the LENGTH command allows us to find the number of runs
## with population size higher than threshold. This is very tricky to follow if you are not careful, because
## it relies on the 'empty' matrix stoch.pop being filled with NA values for missing data, and the BREAK
## command jumping out of the simulation loops when pop <= threshold. The command that calculates
## percentage.under is taking the total number of runs, then subtracting the number of runs that
## stayed above threshold, then dividing this by the number of runs and multiplying by 100 to
## obtain the percentage of runs with pseudo-extinction. When the entry in stoch.pop is greater than threshold,
## the original NA value is replaced with a number (any number, so we can use >=0, but it would also
## work equally well to use >=threshold, because the NA will never be replaced by a population size
## less than threshold).
#################
############### MEAN YEAR WHEN POPULATION SIZE WENT BELOW THRESHOLD, FOR THOSE RUNS WITH A PSEUDO-EXTINCTION
time<-NULL ## create an empty vector to hold results
for (i in 1:runs){ ## loop through all the iterations of the simulation
t<-max(which(stoch.pop[,i]>0)) ## Find the maximum time when Population was >0 (as opposed to remaining NA) for each run
time<-c(time,t) ## Add this value to the end of a vector storing the times
} ## end the loop
time.under<-time[which(time<68)]
hist(time.under,nclass=20)
abline(v=median(time.under),lw=3)
#################
##################################### Environmental stochasticity
project=68
runs=1000
stoch.pop=matrix(NA,project,runs)
## Divide r empirical data into "bad years (r<0)" and "good years (r>0)"
geomean.bad<-mean(r[which(r<0)])
geomean.good<-mean(r[which(r>0)])
stoch.pop[1,]=N0
# loop to create stochastic population sizes
for (i in 1:runs){ # looping over 1000 runs of the stochastic model
for (t in 2:project){ # and looping for 68 years of projection
geomean=sample(x=c(geomean.bad,geomean.good),size=1,prob=c(0.5,0.5),replace=T) #Sample 1 number form vector x with probabilities prob
lambda=exp(rnorm(1,geomean,sd.r)) # draw a value of lambda from a lognormal distribution
stoch.pop[t,i]=stoch.pop[(t-1),i]*lambda # and project one time step from the current pop size
if(stoch.pop[t,i]<=threshold) break # leave the loop if pop <= threshold
}
}
# examine the stochastic output (note that this is huge, 68 x 1000)
stoch.pop
percentage.under<-(runs-length(which(stoch.pop[project,]>=0)))/runs*100 ## WHICH command allows to find the arrow number in the vector which values is higher than 0 (as opposite to be NA))
percentage.under
########################################################
# Plot the projected results
par(mfrow=c(2,1))
stoch.pop.mean=apply(stoch.pop,1,mean, na.rm=T)
log.pop.sd =apply(log(stoch.pop),1,sd, na.rm=T)
ucl =exp(log(stoch.pop.mean)+1.96*log.pop.sd)
lcl =exp(log(stoch.pop.mean)-1.96*log.pop.sd)
plot(1:project,stoch.pop.mean,'b',pch = 19, col = 'blue', ylim=c(0,max(ucl)),xlab='Years from 1959', ylab='Population Size')
lines(1:project,lcl,'l', col = 'blue')
lines(1:project,ucl,'l', col = 'blue')
#add the original data to evaluate hindcasting
points(1:length(year), N, pch = 25, col = 'red', bg = 'red')
legend(0,2100, c("Projection with 95% CL", "Data"), lty = 1, col = c('blue', 'red'), pch =c(19,25),cex=0.7,bt="n")
# same again with log Y axis so that the upper CL does not obscure other patterns
plot(1:project,stoch.pop.mean, log = "y", type = 'b', pch = 19, col = 'blue', ylim=c(1,max(ucl)), xlab='Years from 1959',
ylab='Population Size')
lines(1:project,lcl,'l', col = 'blue')
lines(1:project,ucl,'l', col = 'blue')
points(1:length(year), N, pch = 25, col = 'red', bg = 'red')
legend(0,2000, c("Projection with 95% CL", "Data"), lty = 1, col = c('blue', 'red'), pch =c(19,25),cex=0.7,bt="n")
#################################################
# Extinction Risk using popbio package
library(popbio) # load the popbio package
# stoch.projection is a function defined in the popbio package. As with any
# function, you can type ?stoch.projection to see the arguments for the
# function and its outputs.
# number of projections, i.e. the number of simulated propoualtion growth trajectories
reps = 1000
#set of growth rates to sample with replacement for growth at each time step
lambda=(N[-1]/N[-length(N)])
final.N<-stoch.projection(as.list(lambda), N[1], nreps=reps, tmax = 50)
final.N
par(mfrow=c(2,1))
hist(final.N, xlab = "Population projected for year 50", col = 'light blue', main = "Exponential growth model",nclass=20)
abline(v=sum(n), lty=3) ### ????????????
ext.threshold = 80 # set the definition of pseudoextinction, i.e. the number below which the
# population should be re-listed under the ESA, or some similar threshold of interest
abline(v=ext.threshold, lty=1, col = 'red')
#########################
ext = 0 # zero the counter for number of extinctions before tallying extinctions
# loop to tally up the number of cases where N.final is < ext.threshold
for (i in 1:length(final.N))
{if (final.N[i] < ext.threshold) ext = ext+1
}
ext # number of cases where N.final is < ext.threshold
prob.ext = ext/reps
prob.ext # proportion of cases where N.final is < ext.threshold, i.e. Prob{pseudoextinction}
###########OR = Calculation of treshold
ext<-length(which(final.N