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Sahul megafauna demographic susceptibility-base models.R
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Sahul megafauna demographic susceptibility-base models.R
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##########################################################################################################################################
## megafauna demographic susceptibility model
## AIM: construct plausible stochastic demographic models for main Sahul megafauna to determine relative demographic
## susceptibility to environmental change & novel predation (human) sources
##
## VOMBATIFORM HERBIVORES: ✓Diprotodon (†), ✓Palorchestes (†), ✓Zygomaturus (†), ✓Phascolonus (†), ✓Vombatus ursinus
## MACROPODIFORM HERBIVORES: ✓Protemnodon (†), ✓Osphranter rufus, ✓Sthenurus (†), ✓Simosthenurus (†), ✓Procoptodon (†), ✓Metasthenurus (†), ✓Notamacropus
## LARGE BIRDS: ✓Genyornis (†), ✓Dromaius novaehollandiae, ✓Alectura lathami
## CARNIVORES: ✓Sarcophilus, ✓Thylacinus (†), ✓Thylacoleo (†), ✓Dasyurus
## MONOTREMES: ✓Megalibgwilia (†), ✓Tachyglossus
##
## Corey Bradshaw
## corey.bradshaw@flinders.edu.au
## Flinders University, August 2020
##########################################################################################################################################
## functions
# beta distribution shape parameter estimator function
estBetaParams <- function(mu, var) {
alpha <- ((1 - mu) / var - 1 / mu) * mu ^ 2
beta <- alpha * (1 / mu - 1)
return(params = list(alpha = alpha, beta = beta))
}
AICc <- function(...) {
models <- list(...)
num.mod <- length(models)
AICcs <- numeric(num.mod)
ns <- numeric(num.mod)
ks <- numeric(num.mod)
AICc.vec <- rep(0,num.mod)
for (i in 1:num.mod) {
if (length(models[[i]]$df.residual) == 0) n <- models[[i]]$dims$N else n <- length(models[[i]]$residuals)
if (length(models[[i]]$df.residual) == 0) k <- sum(models[[i]]$dims$ncol) else k <- (length(models[[i]]$coeff))+1
AICcs[i] <- (-2*logLik(models[[i]])) + ((2*k*n)/(n-k-1))
ns[i] <- n
ks[i] <- k
AICc.vec[i] <- AICcs[i]
}
return(AICc.vec)
}
delta.AIC <- function(x) x - min(x) ## where x is a vector of AIC
weight.AIC <- function(x) (exp(-0.5*x))/sum(exp(-0.5*x)) ## Where x is a vector of dAIC
ch.dev <- function(x) ((( as.numeric(x$null.deviance) - as.numeric(x$deviance) )/ as.numeric(x$null.deviance))*100) ## % change in deviance, where x is glm object
linreg.ER <- function(x,y) { # where x and y are vectors of the same length; calls AICc, delta.AIC, weight.AIC functions
fit.full <- lm(y ~ x); fit.null <- lm(y ~ 1)
AIC.vec <- c(AICc(fit.full),AICc(fit.null))
dAIC.vec <- delta.AIC(AIC.vec); wAIC.vec <- weight.AIC(dAIC.vec)
ER <- wAIC.vec[1]/wAIC.vec[2]
r.sq.adj <- as.numeric(summary(fit.full)[9])
return(c(ER,r.sq.adj))
}
## source
source("matrixOperators.r")
# MACROPOD CORRECTIONS (data from Fisher et al. 2001. The ecological basis of life history variation in marsupials. Ecology 82:3531-3540. doi:10.1890/0012-9658(2001)082[3531:TEBOLH]2.0.CO;2)
# mass
NR.mass <- 5.1 # Notamacropus rufogriseus
OR.mass <- 25 # Osphranter rufus
DOL.mass <- 3.57 # Dorcopsis luctuosa
LC.mass <- 3 # Lagorchestes conspicillatus
LH.mass <- 1.3 # L. hirsutus
LF.mass <- 1.8 # L. fasciatus
MAg.mass <- 11 # Macropus agilis
MAn.mass <- 17.5 # M. antilopus
MDo.mass <- 6.5 # M. dorsalis
MFu.mass <- 16 # M. fuliginosus
MGi.mass <- 17.8 # M. giganteus
MPa.mass <- 3.55 # M. parma
MParr.mass <- 11 # M. parryi
MRo.mass <- 15.6 # M. robustus
PAs.mass <- 4.3 # Petrogale assimilis
PI.mass <- 4.2 # P. inomata
PP.mass <- 6.3 # P. penicillata
PPe.mass <- 5.2 # P. persephone
PX.mass <- 7 # P. xanthopus
TB.mass <- 3.9 # Thylogale billardienii
TS.mass <- 4.1 # T. stigmatica
TT.mass <- 3.8 # T. thetis
WB.mass <- 13.0 # Wallabia bicolor
# inter-birth interval
NR.IBI <- 286
OR.IBI <- 241
DOL.IBI <- 191
LC.IBI <- 153
LH.IBI <- 125
LF.IBI <- 365
MAg.IBI <- 220
MAn.IBI <- 270
MDo.IBI <- 211
MFu.IBI <- 372
MGi.IBI <- 363
MPa.IBI <- 213
MParr.IBI <- 266
MRo.IBI <- mean(c(256,264))
PAs.IBI <- 200
PI.IBI <- 210
PP.IBI <- 205
PPe.IBI <- 209
PX.IBI <- 196
TB.IBI <- 204
TS.IBI <- 185
TT.IBI <- 182
WB.IBI <- 256
MACROPOD.IBI <- c(NR.IBI,OR.IBI,DOL.IBI,LC.IBI,LH.IBI,LF.IBI,MAg.IBI,MAn.IBI,MDo.IBI,MFu.IBI,MGi.IBI,MPa.IBI,MParr.IBI,MRo.IBI,PAs.IBI,PI.IBI,PP.IBI,PPe.IBI,PX.IBI,TB.IBI,TS.IBI,TT.IBI,WB.IBI)
MACROPOD.mass <- c(NR.mass,OR.mass,DOL.mass,LC.mass,LH.mass,LF.mass,MAg.mass,MAn.mass,MDo.mass,MFu.mass,MGi.mass,MPa.mass,MParr.mass,MRo.mass,PAs.mass,PI.mass,PP.mass,PPe.mass,PX.mass,TB.mass,TS.mass,TT.mass,WB.mass)
plot(log10(MACROPOD.mass), MACROPOD.IBI, pch=19)
linreg.ER(log10(MACROPOD.mass), MACROPOD.IBI)
IBImass.fit <- lm(MACROPOD.IBI ~ log10(MACROPOD.mass))
summary(IBImass.fit)
abline(h=mean(MACROPOD.IBI), lty=2)
abline(IBImass.fit, lty=2, col="red")
MACROPOD.F.corr1 <- 365/mean(MACROPOD.IBI)
MACROPOD.F.corr1
MACROPOD.F.corr.a <- coef(IBImass.fit)[1]
MACROPOD.F.corr.b <- coef(IBImass.fit)[2]
# age at first breeding
NR.alpha <- mean(c(412,420,390))
OR.alpha <- mean(c(913,613))
DOL.alpha <- 450
LC.alpha <- 363
LH.alpha <- mean(c(315,345))
LF.alpha <- 365
MAg.alpha <- mean(c(405,357))
MAn.alpha <- 802
MDo.alpha <- 420
MFu.alpha <- mean(c(604,420))
MGi.alpha <- mean(c(660,600))
MPa.alpha <- 424
MParr.alpha <- mean(c(630,586))
MRo.alpha <- mean(c(535,613,909))
PAs.alpha <- 525
PI.alpha <- 540
PP.alpha <- 540
PPe.alpha <- 683
PX.alpha <- 541
TB.alpha <- 392
TS.alpha <- mean(c(341,336))
TT.alpha <- mean(c(510,600))
WB.alpha <- 450
# predicted age at first breeding
NR.alpha.pred <- (exp(-1.34 + (0.214*log(NR.mass*1000))))
OR.alpha.pred <- (exp(-1.34 + (0.214*log(OR.mass*1000))))
DOL.alpha.pred <- (exp(-1.34 + (0.214*log(DOL.mass*1000))))
LC.alpha.pred <- (exp(-1.34 + (0.214*log(LC.mass*1000))))
LH.alpha.pred <- (exp(-1.34 + (0.214*log(LH.mass*1000))))
LF.alpha.pred <- (exp(-1.34 + (0.214*log(LF.mass*1000))))
MAg.alpha.pred <- (exp(-1.34 + (0.214*log(MAg.mass*1000))))
MAn.alpha.pred <- (exp(-1.34 + (0.214*log(MAn.mass*1000))))
MDo.alpha.pred <- (exp(-1.34 + (0.214*log(MDo.mass*1000))))
MFu.alpha.pred <- (exp(-1.34 + (0.214*log(MFu.mass*1000))))
MGi.alpha.pred <- (exp(-1.34 + (0.214*log(MGi.mass*1000))))
MPa.alpha.pred <- (exp(-1.34 + (0.214*log(MPa.mass*1000))))
MParr.alpha.pred <- (exp(-1.34 + (0.214*log(MParr.mass*1000))))
MRo.alpha.pred <- (exp(-1.34 + (0.214*log(MRo.mass*1000))))
PAs.alpha.pred <- (exp(-1.34 + (0.214*log(PAs.mass*1000))))
PI.alpha.pred <- (exp(-1.34 + (0.214*log(PI.mass*1000))))
PP.alpha.pred <- (exp(-1.34 + (0.214*log(PP.mass*1000))))
PPe.alpha.pred <- (exp(-1.34 + (0.214*log(PPe.mass*1000))))
PX.alpha.pred <- (exp(-1.34 + (0.214*log(PX.mass*1000))))
TB.alpha.pred <- (exp(-1.34 + (0.214*log(TB.mass*1000))))
TS.alpha.pred <- (exp(-1.34 + (0.214*log(TS.mass*1000))))
TT.alpha.pred <- (exp(-1.34 + (0.214*log(TT.mass*1000))))
WB.alpha.pred <- (exp(-1.34 + (0.214*log(WB.mass*1000))))
MACROPOD.alpha.vec <- c(NR.alpha,OR.alpha,DOL.alpha,LC.alpha,LH.alpha,LF.alpha,MAg.alpha,MAn.alpha,MDo.alpha,MFu.alpha,MGi.alpha,MPa.alpha,MParr.alpha,MRo.alpha,PAs.alpha,PI.alpha,PP.alpha,PPe.alpha,PX.alpha,TB.alpha,TS.alpha,TT.alpha,WB.alpha)
MACROPOD.alpha.pred.vec <- c(NR.alpha.pred,OR.alpha.pred,DOL.alpha.pred,LC.alpha.pred,LH.alpha.pred,LF.alpha.pred,MAg.alpha.pred,MAn.alpha.pred,MDo.alpha.pred,MFu.alpha.pred,MGi.alpha.pred,MPa.alpha.pred,MParr.alpha.pred,MRo.alpha.pred,PAs.alpha.pred,PI.alpha.pred,PP.alpha.pred,PPe.alpha.pred,PX.alpha.pred,TB.alpha.pred,TS.alpha.pred,TT.alpha.pred,WB.alpha.pred)
MACROPOD.alpha.vec.yr <- MACROPOD.alpha.vec/365
mean((MACROPOD.alpha.pred.vec - MACROPOD.alpha.vec.yr) / MACROPOD.alpha.pred.vec)
MACROPOD.alpha.corr <- mean(MACROPOD.alpha.vec)/365 / mean(MACROPOD.alpha.pred.vec)
MACROPOD.alpha.corr
plot(MACROPOD.mass, MACROPOD.alpha.vec, pch=19)
# VOMBATIFORM CORRECTIONS
# mass
PC.mass <- 5.1
VU.mass <- 25
LL.mass <- 26
LK.mass <- 31
# inter-birth interval
PC.IBI <- 383 # koala interbirth interval (Martin, R.W. & Handasyde, K.A. 1995 Koala In The Australian Museum complete book of Australian Mammals (ed. R. Strahan) pp.196-198. Sydney: Reed Books)
VU.IBI <- 730 # Vombatus ursinus
LL.IBI <- 365 # southern hairy-nosed wombat
LK.IBI <- 548 # northern hairy-nosed wombat
# age at first breeding (from Fisher et al. 2001)
PC.alpha <- 2
VU.alpha <- 2
LL.alpha <- round(mean(c(1095,540))/365, 0)
# predicted age at first breeding
PC.alpha.pred <- ceiling(exp(-1.34 + (0.214*log(PC.mass*1000))))
VU.alpha.pred <- ceiling(exp(-1.34 + (0.214*log(VU.mass*1000))))
LL.alpha.pred <- ceiling(exp(-1.34 + (0.214*log(LL.mass*1000))))
VOMBAT.alpha.corr <- mean(c(PC.alpha,VU.alpha,LL.alpha)) / mean(c(PC.alpha.pred,VU.alpha.pred,LL.alpha.pred))
VOMBAT.alpha.corr
VOMBAT.IBI <- c(PC.IBI,VU.IBI,LL.IBI,LK.IBI)
VOMBAT.mass <- c(PC.mass,VU.mass,LL.mass,LK.mass)
plot((VOMBAT.mass), VOMBAT.IBI, pch=19)
abline(h=mean(VOMBAT.IBI), lty=2)
VOMBAT.F.corr <- 365/mean(VOMBAT.IBI)
VOMBAT.F.corr
# use macropod relationship to project vombatiforms
mass.range.vec <- seq(1,DP.mass,1)
VOMBAT.IBI.pred <- mean(VOMBAT.IBI) + (MACROPOD.F.corr.b * log10(mass.range.vec))
plot(log10(mass.range.vec),VOMBAT.IBI.pred, type="l", ylim=c(min(VOMBAT.IBI),max(VOMBAT.IBI.pred)))
abline(h=mean(VOMBAT.IBI), lty=2)
points(log10(VOMBAT.mass), VOMBAT.IBI, pch=19)
VOMBAT.EXT.mass <- c(DP.mass, PA.mass, ZT.mass, PH.mass)
VOMBAT.IBI.EXT.pred <- mean(VOMBAT.IBI) + (MACROPOD.F.corr.b * log10(VOMBAT.EXT.mass))
points(log10(VOMBAT.EXT.mass), VOMBAT.IBI.EXT.pred, pch=19, col="red")
# as above, but anchor relationship to VU
VU.IBI.pred <- as.numeric(mean(VOMBAT.IBI) + (MACROPOD.F.corr.b * log10(VU.mass)))
VOMBAT.IBI.pred2 <- (mean(VOMBAT.IBI) + (VU.IBI - VU.IBI.pred)) + (MACROPOD.F.corr.b * log10(mass.range.vec))
plot(log10(mass.range.vec),VOMBAT.IBI.pred2, type="l", ylim=c(min(VOMBAT.IBI),max(VOMBAT.IBI.pred2)))
points(log10(VOMBAT.mass), VOMBAT.IBI, pch=19)
VOMBAT.IBI.EXT.pred2 <- (mean(VOMBAT.IBI) + (VU.IBI - VU.IBI.pred)) + (MACROPOD.F.corr.b * log10(VOMBAT.EXT.mass))
points(log10(VOMBAT.EXT.mass), VOMBAT.IBI.EXT.pred2, pch=19, col="red")
## BASE MODELS
############################
## DIPROTODON (optatum) (DP)
## sources: Brook & Johnson 2006 (Alcheringa 30:39-48, http://doi.org/10.1080/03115510609506854)
# mass
DP.mass <- 2786 # kg (Wroe et al. 2004 PRSB 271:S34-S36)
## predicted rm (from Henneman 1983 Oecologia 56:104-108)
## log10rm = 0.6914 - 0.2622*log10m (mass in g)
DP.rm.pred <- 10^(0.6914 - (0.2622*log10(DP.mass*1000)))
DP.lm.pred <- exp(DP.rm.pred)
## theoretical population density for mammalian herbivores based on body size (Damuth 1981; Freeland 1990)
## log10D = 4.196 − 0.74*(log10m)
DP.D.pred <- (10^(4.196 - (0.74*log10(DP.mass*1000))))/2 # divided by 2 for females only
DP.D.pred # animals/km2
## max age
## non-volant birds & mammals (Healy K et al. 2014 PRSB)
## log10ls = 0.89 + 0.13log10m (mass in grams; ls = years)
DP.age.max1 <- round(10^(0.89 + (0.13*log10(DP.mass*1000))), 0)
DP.age.max <- round(DP.age.max1 * 26/29, 0) # corrected for over-estimate derived from Vombatus
## age vector
DP.age.vec <- 0:DP.age.max
## fertility
## total fecundity from Allainé et al. 1987 (Oecologia)
## lnF = 2.719 - 0.211lnM (all mammals)
DP.F.pred1 <- exp(2.719 - (0.211*log(DP.mass*1000)))/2 # divided by 2 for females
DP.F.pred <- DP.F.pred1 * (365/as.numeric((mean(VOMBAT.IBI) + (VU.IBI - VU.IBI.pred)) + (MACROPOD.F.corr.b * log10(DP.mass))))
## age at primiparity
## lnalpha = 0.214 + 0.263*lnM (https://dx.doi.org/10.1093%2Fgerona%2F62.2.149)
DP.alpha1 <- ceiling(exp(-1.34 + (0.214*log(DP.mass*1000))))
DP.alpha <- DP.alpha1
## define m function with age
DP.m.vec <- c(rep(0, DP.alpha-1), rep(0.75*DP.F.pred, round(DP.alpha/2,0)), rep(DP.F.pred, (DP.age.max+1-((DP.alpha-1+round(DP.alpha/2,0))))))
DP.m.sd.vec <- 0.05*DP.m.vec
plot(DP.age.vec, DP.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
# fit sigmoidal function
# logistic power function y = a / (1+(x/b)^c)
DP.m.dat <- data.frame(DP.age.vec, DP.m.vec)
param.init <- c(0.3, 6, -5)
DP.fit.logp <- nls(DP.m.vec ~ a / (1+(DP.age.vec/b)^c),
data = DP.m.dat,
algorithm = "port",
start = c(a = param.init[1], b = param.init[2], c = param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
DP.fit.logp.summ <- summary(DP.fit.logp)
plot(DP.age.vec, DP.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
DP.age.vec.cont <- seq(0,max(DP.age.vec),1)
DP.pred.p.m <- coef(DP.fit.logp)[1] / (1+(DP.age.vec.cont/coef(DP.fit.logp)[2])^coef(DP.fit.logp)[3])
DP.pred.p.mm <- ifelse(DP.pred.p.m > 1, 1, DP.pred.p.m)
lines(DP.age.vec.cont, DP.pred.p.mm,lty=2,lwd=3,col="red")
## survival
## mean adult survival (McCarthy et al. 2008 Am Nat)
## ln{-ln[s(t)]} = ln(a) + bln(M) + ln (t)
ln.a.s <- -0.5; b.s <- -0.25
DP.s.tran <- ln.a.s + b.s*log(DP.mass*1000) + log(1)
DP.s.ad.yr <- exp(-exp(DP.s.tran))
# Siler hazard h(x) (Gurven et al. 2007)
a1 <- 1 - (1*DP.s.ad.yr) # initial infant mortality rate (also known as αt)
b1 <- 1.0 # rate of mortality decline (also known as bt)
a2 <- 1 - DP.s.ad.yr # age-independent mortality (exogenous mortality due to environment); also known as ct
a3 <- 0.9e-04 # initial adult mortality rate (also known as βt)
b3 <- 0.05 # rate of mortality increase
longev <- DP.age.max
x <- seq(0,longev,1) # age vector
h.x <- a1 * exp(-b1*x) + a2 + a3 * exp(b3 * x) # Siler's hazard model
plot(x,h.x,pch=19,type="l")
plot(x,log(h.x),pch=19,type="l")
l.x <- exp((-a1/b1) * (1 - exp(-b1*x))) * exp(-a2 * x) * exp(a3/b3 * (1 - exp(b3 * x))) # Siler's survival (proportion surviving) model
init.pop <- 10000
lx <- round(init.pop*l.x,0)
len.lx <- length(lx)
dx <- lx[1:(len.lx-1)]-lx[2:len.lx]
qx <- dx/lx[1:(length(lx)-1)]
DP.Sx <- c(0.99*DP.s.ad.yr, 1 - qx)
plot(x, DP.Sx, pch=19, type="l", xlab="age (years)", ylab="Sx")
DP.s.sd.vec <- 0.05*DP.Sx
## create matrix
DP.popmat <- matrix(data = 0, nrow=DP.age.max+1, ncol=DP.age.max+1)
diag(DP.popmat[2:(DP.age.max+1),]) <- DP.Sx[-(DP.age.max+1)]
DP.popmat[DP.age.max+1,DP.age.max+1] <- DP.Sx[DP.age.max+1]
DP.popmat[1,] <- DP.pred.p.mm
colnames(DP.popmat) <- c(0:DP.age.max)
rownames(DP.popmat) <- c(0:DP.age.max)
DP.popmat.orig <- DP.popmat ## save original matrix
## matrix properties
max.lambda(DP.popmat.orig) ## 1-yr lambda
DP.lm.pred
max.r(DP.popmat.orig) # rate of population change, 1-yr
DP.ssd <- stable.stage.dist(DP.popmat.orig) ## stable stage distribution
plot(DP.age.vec, DP.ssd, type="l", pch=19, xlab="age (yrs)", ylab="ssd")
R.val(DP.popmat.orig, DP.age.max) # reproductive value
DP.gen.l <- G.val(DP.popmat.orig, DP.age.max) # mean generation length
## initial population vector
area <- 500*500 # km × km = 250,000 km^2; equates to approximatley 10% larger than State of Victoria (227,444 km^2)
DP.pop.found <- round(area*DP.D.pred, 0) # founding population size (estimated density * 100 × 100 km region [10,000 km2])
DP.init.vec <- DP.ssd * DP.pop.found
#################
## project
## set time limit for projection in 1-yr increments
yr.st <- 1
#************************
yr.end <- round(40*DP.gen.l, 0) # set projection end date
#************************
t <- (yr.end - yr.st)
DP.tot.F <- sum(DP.popmat.orig[1,])
DP.popmat <- DP.popmat.orig
yr.vec <- seq(yr.st,yr.end)
## set population storage matrices
DP.n.mat <- matrix(0, nrow=DP.age.max+1,ncol=(t+1))
DP.n.mat[,1] <- DP.init.vec
## set up projection loop
for (i in 1:t) {
DP.n.mat[,i+1] <- DP.popmat %*% DP.n.mat[,i]
}
DP.n.pred <- colSums(DP.n.mat)
yrs <- seq(yr.st, yr.end, 1)
plot(yrs, log10(DP.n.pred),type="l",lty=2,pch=19,xlab="year",ylab="log10 N")
# compensatory density feedback
DP.K.max <- 1*DP.pop.found
DP.K.vec <- c(1, DP.K.max/2, 0.75*DP.K.max, DP.K.max)
DP.red.vec <- c(1,0.98,0.96,0.9383)
plot(DP.K.vec, DP.red.vec,pch=19,type="b")
DP.Kred.dat <- data.frame(DP.K.vec, DP.red.vec)
# logistic power function a/(1+(x/b)^c)
DP.param.init <- c(1, 2*DP.K.max, 2)
DP.fit.lp <- nls(DP.red.vec ~ a/(1+(DP.K.vec/b)^c),
data = DP.Kred.dat,
algorithm = "port",
start = c(a = DP.param.init[1], b = DP.param.init[2], c = DP.param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
DP.fit.lp.summ <- summary(DP.fit.lp)
plot(DP.K.vec, DP.red.vec, pch=19,xlab="N",ylab="reduction factor")
DP.K.vec.cont <- seq(1,2*DP.pop.found,1)
DP.pred.lp.fx <- coef(DP.fit.lp)[1]/(1+(DP.K.vec.cont/coef(DP.fit.lp)[2])^coef(DP.fit.lp)[3])
lines(DP.K.vec.cont, DP.pred.lp.fx, lty=3,lwd=3,col="red")
DP.a.lp <- coef(DP.fit.lp)[1]
DP.b.lp <- coef(DP.fit.lp)[2]
DP.c.lp <- coef(DP.fit.lp)[3]
## compensatory density-feedback deterministic model
## set population storage matrices
DP.n.mat <- matrix(0, nrow=DP.age.max+1, ncol=(t+1))
DP.n.mat[,1] <- DP.init.vec
DP.popmat <- DP.popmat.orig
## set up projection loop
for (i in 1:t) {
DP.totN.i <- sum(DP.n.mat[,i])
DP.pred.red <- as.numeric(DP.a.lp/(1+(DP.totN.i/DP.b.lp)^DP.c.lp))
diag(DP.popmat[2:(DP.age.max+1),]) <- (DP.Sx[-(DP.age.max+1)])*DP.pred.red
DP.popmat[DP.age.max+1,DP.age.max+1] <- (DP.Sx[DP.age.max+1])*DP.pred.red
DP.popmat[1,] <- DP.pred.p.mm
DP.n.mat[,i+1] <- DP.popmat %*% DP.n.mat[,i]
}
DP.n.pred <- colSums(DP.n.mat)
plot(yrs, DP.n.pred, type="l",lty=2,pch=19,xlab="year",ylab="N")
abline(h=DP.pop.found, lty=2, col="red", lwd=2)
## stochatic projection with density feedback
## set storage matrices & vectors
iter <- 100
itdiv <- iter/10
DP.n.sums.mat <- matrix(data=NA, nrow=iter, ncol=(t+1))
DP.s.arr <- DP.m.arr <- array(data=NA, dim=c(t+1, DP.age.max+1, iter))
for (e in 1:iter) {
DP.popmat <- DP.popmat.orig
DP.n.mat <- DP.s.mat <- DP.m.mat <- matrix(0, nrow=DP.age.max+1,ncol=(t+1))
DP.n.mat[,1] <- DP.init.vec
DP.s.mat[,1] <- DP.Sx
DP.m.mat[,1] <- DP.pred.p.mm
for (i in 1:t) {
# stochastic survival values
DP.s.alpha <- estBetaParams(DP.Sx, DP.s.sd.vec^2)$alpha
DP.s.beta <- estBetaParams(DP.Sx, DP.s.sd.vec^2)$beta
DP.s.stoch <- rbeta(length(DP.s.alpha), DP.s.alpha, DP.s.beta)
# stochastic fertilty sampler (gaussian)
DP.fert.stch <- rnorm(length(DP.popmat[,1]), DP.pred.p.mm, DP.m.sd.vec)
DP.m.arr[i,,e] <- ifelse(DP.fert.stch < 0, 0, DP.fert.stch)
DP.totN.i <- sum(DP.n.mat[,i], na.rm=T)
DP.pred.red <- DP.a.lp/(1+(DP.totN.i/DP.b.lp)^DP.c.lp)
diag(DP.popmat[2:(DP.age.max+1),]) <- (DP.s.stoch[-(DP.age.max+1)])*DP.pred.red
DP.popmat[DP.age.max+1,DP.age.max+1] <- (DP.s.stoch[DP.age.max+1])*DP.pred.red
DP.popmat[1,] <- DP.m.arr[i,,e]
DP.n.mat[,i+1] <- DP.popmat %*% DP.n.mat[,i]
DP.s.arr[i,,e] <- DP.s.stoch * DP.pred.red
} # end i loop
DP.n.sums.mat[e,] <- ((as.vector(colSums(DP.n.mat))/DP.pop.found))
if (e %% itdiv==0) print(e)
} # end e loop
DP.n.md <- apply(DP.n.sums.mat, MARGIN=2, median, na.rm=T) # mean over all iterations
DP.n.up <- apply(DP.n.sums.mat, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
DP.n.lo <- apply(DP.n.sums.mat, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
par(mfrow=c(1,3))
plot(yrs,DP.n.md,type="l", main = "", xlab="year", ylab="pN1", lwd=2, ylim=c(0.95*min(DP.n.lo),1.05*max(DP.n.up)))
lines(yrs,DP.n.lo,lty=2,col="red",lwd=1.5)
lines(yrs,DP.n.up,lty=2,col="red",lwd=1.5)
DP.s.add <- DP.m.add <- rep(0, DP.age.max+1)
for (m in 1:iter) {
DP.s.add <- rbind(DP.s.add, DP.s.arr[ceiling(DP.gen.l):(t+1),,m])
DP.m.add <- rbind(DP.m.add, DP.m.arr[ceiling(DP.gen.l):(t+1),,m])
}
DP.s.add <- DP.s.add[-1,]
DP.m.add <- DP.m.add[-1,]
DP.s.md <- apply(DP.s.add, MARGIN=2, median, na.rm=T) # mean s over all iterations
DP.s.up <- apply(DP.s.add, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
DP.s.lo <- apply(DP.s.add, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
plot(DP.age.vec,DP.s.md,type="l", main = "", xlab="age", ylab="s", lwd=2, ylim=c(0.95*min(DP.s.lo),1.05*max(DP.s.up)))
lines(DP.age.vec,DP.s.lo,lty=2,col="red",lwd=1.5)
lines(DP.age.vec,DP.s.up,lty=2,col="red",lwd=1.5)
DP.m.md <- apply(DP.m.add, MARGIN=2, median, na.rm=T) # mean s over all iterations
DP.m.up <- apply(DP.m.add, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
DP.m.lo <- apply(DP.m.add, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
plot(DP.age.vec,DP.m.md,type="l", main = "", xlab="age", ylab="m", lwd=2, ylim=c(0.95*min(DP.m.lo),1.05*max(DP.m.up)))
lines(DP.age.vec,DP.m.lo,lty=2,col="red",lwd=1.5)
lines(DP.age.vec,DP.m.up,lty=2,col="red",lwd=1.5)
par(mfrow=c(1,1))
############################
## PALORCHESTES (azael) (PA)
## sources: Brook & Johnson 2006 (Alcheringa 30:39-48, http://doi.org/10.1080/03115510609506854)
## Richards et al. 2019 (PLoS One https://doi.org/10.1371/journal.pone.0221824)
# mass
PA.mass <- 1000 # kg (Richards et al. 2019 (PLoS One https://doi.org/10.1371/journal.pone.0221824))
## predicted rm (from Henneman 1983 Oecologia 56:104-108)
## log10rm = 0.6914 - 0.2622*log10m (mass in g)
PA.rm.pred <- 10^(0.6914 - (0.2622*log10(PA.mass*1000)))
PA.lm.pred <- exp(PA.rm.pred)
## theoretical population density for mammalian herbivores based on body size (Damuth 1981; Freeland 1990)
## log10D = 4.196 − 0.74*(log10m)
PA.D.pred <- (10^(4.196 - (0.74*log10(PA.mass*1000))))/2 # divided by 2 for females only
PA.D.pred # animals/km2
## max age
## non-volant birds & mammals (Healy K et al. 2014 PRSB)
## log10ls = 0.89 + 0.13log10m (mass in grams; ls = years)
PA.age.max1 <- round(10^(0.89 + (0.13*log10(PA.mass*1000))), 0)
PA.age.max <- round(PA.age.max1 * 26/29, 0) # corrected for over-estimate derived from Vombatus
## age vector
PA.age.vec <- 0:PA.age.max
## fertility
## total fecundity from Allainé et al. 1987 (Oecologia)
## lnF = 2.719 - 0.211lnM (all mammals)
PA.F.pred1 <- exp(2.719 - (0.211*log(PA.mass*1000)))/2 # divided by 2 for females
PA.F.pred <- PA.F.pred1 * (365/as.numeric((mean(VOMBAT.IBI) + (VU.IBI - VU.IBI.pred)) + (MACROPOD.F.corr.b * log10(PA.mass))))
## age at primiparity
## lnalpha = 0.214 + 0.263*lnM (https://dx.doi.org/10.1093%2Fgerona%2F62.2.149)
PA.alpha1 <- ceiling(exp(-1.34 + (0.214*log(PA.mass*1000))))
PA.alpha <- PA.alpha1
## define m function with age
PA.m.vec <- c(rep(0, PA.alpha-1), rep(0.75*PA.F.pred, round(PA.alpha/2,0)), rep(PA.F.pred, (PA.age.max+1-((PA.alpha-1+round(PA.alpha/2,0))))))
PA.m.sd.vec <- 0.05*PA.m.vec
plot(PA.age.vec, PA.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
# fit sigmoidal function
# logistic power function y = a / (1+(x/b)^c)
PA.m.dat <- data.frame(PA.age.vec, PA.m.vec)
param.init <- c(0.3, 7, -5)
PA.fit.logp <- nls(PA.m.vec ~ a / (1+(PA.age.vec/b)^c),
data = PA.m.dat,
algorithm = "port",
start = c(a = param.init[1], b = param.init[2], c = param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
PA.fit.logp.summ <- summary(PA.fit.logp)
plot(PA.age.vec, PA.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
PA.age.vec.cont <- seq(0,max(PA.age.vec),1)
PA.pred.p.m <- coef(PA.fit.logp)[1] / (1+(PA.age.vec.cont/coef(PA.fit.logp)[2])^coef(PA.fit.logp)[3])
PA.pred.p.mm <- ifelse(PA.pred.p.m > 1, 1, PA.pred.p.m)
lines(PA.age.vec.cont, PA.pred.p.mm,lty=2,lwd=3,col="red")
## survival
## mean adult survival (McCarthy et al. 2008 Am Nat)
## ln{-ln[s(t)]} = ln(a) + bln(M) + ln (t)
ln.a.s <- -0.5; b.s <- -0.25
PA.s.tran <- ln.a.s + b.s*log(PA.mass*1000) + log(1)
PA.s.ad.yr <- exp(-exp(PA.s.tran))
# Siler hazard h(x) (Gurven et al. 2007)
a1 <- 1 - (PA.s.ad.yr) # initial infant mortality rate (also known as αt)
b1 <- 1.5 # rate of mortality decline (also known as bt)
a2 <- 1 - PA.s.ad.yr # age-independent mortality (exogenous mortality due to environment); also known as ct
a3 <- 0.7e-04 # initial adult mortality rate (also known as βt)
b3 <- 0.05 # rate of mortality increase
longev <- PA.age.max
x <- seq(0,longev,1) # age vector
h.x <- a1 * exp(-b1*x) + a2 + a3 * exp(b3 * x) # Siler's hazard model
plot(x,h.x,pch=19,type="l")
plot(x,log(h.x),pch=19,type="l")
l.x <- exp((-a1/b1) * (1 - exp(-b1*x))) * exp(-a2 * x) * exp(a3/b3 * (1 - exp(b3 * x))) # Siler's survival (proportion surviving) model
init.pop <- 10000
lx <- round(init.pop*l.x,0)
len.lx <- length(lx)
dx <- lx[1:(len.lx-1)]-lx[2:len.lx]
qx <- dx/lx[1:(length(lx)-1)]
PA.Sx <- c(0.99*PA.s.ad.yr, 1 - qx)
plot(x, PA.Sx, pch=19, type="l", xlab="age (years)", ylab="Sx")
PA.s.sd.vec <- 0.05*PA.Sx
## create matrix
PA.popmat <- matrix(data = 0, nrow=PA.age.max+1, ncol=PA.age.max+1)
diag(PA.popmat[2:(PA.age.max+1),]) <- PA.Sx[-(PA.age.max+1)]
PA.popmat[PA.age.max+1,PA.age.max+1] <- PA.Sx[PA.age.max+1]
PA.popmat[1,] <- PA.pred.p.mm
colnames(PA.popmat) <- c(0:PA.age.max)
rownames(PA.popmat) <- c(0:PA.age.max)
PA.popmat.orig <- PA.popmat ## save original matrix
## matrix properties
max.lambda(PA.popmat.orig) ## 1-yr lambda
PA.lm.pred
max.r(PA.popmat.orig) # rate of population change, 1-yr
PA.ssd <- stable.stage.dist(PA.popmat.orig) ## stable stage distribution
plot(PA.age.vec, PA.ssd, type="l", pch=19, xlab="age (yrs)", ylab="ssd")
R.val(PA.popmat.orig, PA.age.max) # reproductive value
PA.gen.l <- G.val(PA.popmat.orig, PA.age.max) # mean generation length
## initial population vector
area <- 500*500 # km × km
PA.pop.found <- round(area*PA.D.pred, 0) # founding population size (estimated density * 100 × 100 km region [10,000 km2])
PA.init.vec <- PA.ssd * PA.pop.found
#################
## project
## set time limit for projection in 1-yr increments
yr.st <- 1
#************************
yr.end <- round(40*PA.gen.l, 0) # set projection end date
#************************
t <- (yr.end - yr.st)
PA.tot.F <- sum(PA.popmat.orig[1,])
PA.popmat <- PA.popmat.orig
yr.vec <- seq(yr.st,yr.end)
## set population storage matrices
PA.n.mat <- matrix(0, nrow=PA.age.max+1,ncol=(t+1))
PA.n.mat[,1] <- PA.init.vec
## set up projection loop
for (i in 1:t) {
PA.n.mat[,i+1] <- PA.popmat %*% PA.n.mat[,i]
}
PA.n.pred <- colSums(PA.n.mat)
yrs <- seq(yr.st, yr.end, 1)
plot(yrs, log10(PA.n.pred),type="l",lty=2,pch=19,xlab="year",ylab="log10 N")
# compensatory density feedback
PA.K.max <- 1*PA.pop.found
PA.K.vec <- c(1, PA.K.max/2, 0.75*PA.K.max, PA.K.max)
PA.red.vec <- c(1,0.982,0.957,0.9206)
plot(PA.K.vec, PA.red.vec,pch=19,type="b")
PA.Kred.dat <- data.frame(PA.K.vec, PA.red.vec)
# logistic power function a/(1+(x/b)^c)
PA.param.init <- c(1, 2*PA.K.max, 2)
PA.fit.lp <- nls(PA.red.vec ~ a/(1+(PA.K.vec/b)^c),
data = PA.Kred.dat,
algorithm = "port",
start = c(a = PA.param.init[1], b = PA.param.init[2], c = PA.param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
PA.fit.lp.summ <- summary(PA.fit.lp)
plot(PA.K.vec, PA.red.vec, pch=19,xlab="N",ylab="reduction factor")
PA.K.vec.cont <- seq(1,2*PA.pop.found,1)
PA.pred.lp.fx <- coef(PA.fit.lp)[1]/(1+(PA.K.vec.cont/coef(PA.fit.lp)[2])^coef(PA.fit.lp)[3])
lines(PA.K.vec.cont, PA.pred.lp.fx, lty=3,lwd=3,col="red")
PA.a.lp <- coef(PA.fit.lp)[1]
PA.b.lp <- coef(PA.fit.lp)[2]
PA.c.lp <- coef(PA.fit.lp)[3]
## compensatory density-feedback deterministic model
## set population storage matrices
PA.n.mat <- matrix(0, nrow=PA.age.max+1, ncol=(t+1))
PA.n.mat[,1] <- PA.init.vec
PA.popmat <- PA.popmat.orig
## set up projection loop
for (i in 1:t) {
PA.totN.i <- sum(PA.n.mat[,i])
PA.pred.red <- as.numeric(PA.a.lp/(1+(PA.totN.i/PA.b.lp)^PA.c.lp))
diag(PA.popmat[2:(PA.age.max+1),]) <- (PA.Sx[-(PA.age.max+1)])*PA.pred.red
PA.popmat[PA.age.max+1,PA.age.max+1] <- (PA.Sx[PA.age.max+1])*PA.pred.red
PA.popmat[1,] <- PA.pred.p.mm
PA.n.mat[,i+1] <- PA.popmat %*% PA.n.mat[,i]
}
PA.n.pred <- colSums(PA.n.mat)
plot(yrs, PA.n.pred, type="l",lty=2,pch=19,xlab="year",ylab="N")
abline(h=PA.pop.found, lty=2, col="red", lwd=2)
## stochatic projection with density feedback
## set storage matrices & vectors
iter <- 100
itdiv <- iter/10
PA.n.sums.mat <- matrix(data=NA, nrow=iter, ncol=(t+1))
PA.s.arr <- PA.m.arr <- array(data=NA, dim=c(t+1, PA.age.max+1, iter))
for (e in 1:iter) {
PA.popmat <- PA.popmat.orig
PA.n.mat <- matrix(0, nrow=PA.age.max+1,ncol=(t+1))
PA.n.mat[,1] <- PA.init.vec
for (i in 1:t) {
# stochastic survival values
PA.s.alpha <- estBetaParams(PA.Sx, PA.s.sd.vec^2)$alpha
PA.s.beta <- estBetaParams(PA.Sx, PA.s.sd.vec^2)$beta
PA.s.stoch <- rbeta(length(PA.s.alpha), PA.s.alpha, PA.s.beta)
# stochastic fertilty sampler (gaussian)
PA.fert.stch <- rnorm(length(PA.popmat[,1]), PA.pred.p.mm, PA.m.sd.vec)
PA.m.arr[i,,e] <- ifelse(PA.fert.stch < 0, 0, PA.fert.stch)
PA.totN.i <- sum(PA.n.mat[,i], na.rm=T)
PA.pred.red <- PA.a.lp/(1+(PA.totN.i/PA.b.lp)^PA.c.lp)
diag(PA.popmat[2:(PA.age.max+1),]) <- (PA.s.stoch[-(PA.age.max+1)])*PA.pred.red
PA.popmat[PA.age.max+1,PA.age.max+1] <- (PA.s.stoch[PA.age.max+1])*PA.pred.red
PA.popmat[1,] <- PA.m.arr[i,,e]
PA.n.mat[,i+1] <- PA.popmat %*% PA.n.mat[,i]
PA.s.arr[i,,e] <- PA.s.stoch * PA.pred.red
} # end i loop
PA.n.sums.mat[e,] <- ((as.vector(colSums(PA.n.mat))/PA.pop.found))
if (e %% itdiv==0) print(e)
} # end e loop
PA.n.md <- apply(PA.n.sums.mat, MARGIN=2, median, na.rm=T) # mean over all iterations
PA.n.up <- apply(PA.n.sums.mat, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
PA.n.lo <- apply(PA.n.sums.mat, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
par(mfrow=c(1,3))
plot(yrs,PA.n.md,type="l", main = "", xlab="year", ylab="pN1", lwd=2, ylim=c(0.95*min(PA.n.lo),1.05*max(PA.n.up)))
lines(yrs,PA.n.lo,lty=2,col="red",lwd=1.5)
lines(yrs,PA.n.up,lty=2,col="red",lwd=1.5)
PA.s.add <- PA.m.add <- rep(0, PA.age.max+1)
for (m in 1:iter) {
PA.s.add <- rbind(PA.s.add, PA.s.arr[ceiling(PA.gen.l):(t+1),,m])
PA.m.add <- rbind(PA.m.add, PA.m.arr[ceiling(PA.gen.l):(t+1),,m])
}
PA.s.add <- PA.s.add[-1,]
PA.m.add <- PA.m.add[-1,]
PA.s.md <- apply(PA.s.add, MARGIN=2, median, na.rm=T) # mean s over all iterations
PA.s.up <- apply(PA.s.add, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
PA.s.lo <- apply(PA.s.add, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
plot(PA.age.vec,PA.s.md,type="l", main = "", xlab="age", ylab="s", lwd=2, ylim=c(0.95*min(PA.s.lo),1.05*max(PA.s.up)))
lines(PA.age.vec,PA.s.lo,lty=2,col="red",lwd=1.5)
lines(PA.age.vec,PA.s.up,lty=2,col="red",lwd=1.5)
PA.m.md <- apply(PA.m.add, MARGIN=2, median, na.rm=T) # mean s over all iterations
PA.m.up <- apply(PA.m.add, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
PA.m.lo <- apply(PA.m.add, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
plot(PA.age.vec,PA.m.md,type="l", main = "", xlab="age", ylab="m", lwd=2, ylim=c(0.95*min(PA.m.lo),1.05*max(PA.m.up)))
lines(PA.age.vec,PA.m.lo,lty=2,col="red",lwd=1.5)
lines(PA.age.vec,PA.m.up,lty=2,col="red",lwd=1.5)
par(mfrow=c(1,1))
##############################
## ZYGOMATURUS (trilobus) (ZT)
## sources: Brook & Johnson 2006 (Alcheringa 30:39-48, http://doi.org/10.1080/03115510609506854)
## Richards et al. 2019 (PLoS One https://doi.org/10.1371/journal.pone.0221824)
# mass
ZT.mass <- 500 # kg (Johnson et al. 2006 Australia’s Mammal Extinctions. 278 pp. Cambridge University Press, Melbourne)
## predicted rm (from Henneman 1983 Oecologia 56:104-108)
## log10rm = 0.6914 - 0.2622*log10m (mass in g)
ZT.rm.pred <- 10^(0.6914 - (0.2622*log10(ZT.mass*1000)))
ZT.lm.pred <- exp(ZT.rm.pred)
## theoretical population density for mammalian herbivores based on body size (Damuth 1981; Freeland 1990)
## log10D = 4.196 − 0.74*(log10m)
ZT.D.pred <- (10^(4.196 - (0.74*log10(ZT.mass*1000))))/2 # divided by 2 for females only
ZT.D.pred # animals/km2
## max age
## non-volant birds & mammals (Healy K et al. 2014 PRSB)
## log10ls = 0.89 + 0.13log10m (mass in grams; ls = years)
ZT.age.max1 <- round(10^(0.89 + (0.13*log10(ZT.mass*1000))), 0)
ZT.age.max <- round(ZT.age.max1 * 26/29, 0) # corrected for over-estimate derived from Vombatus
## age vector
ZT.age.vec <- 0:ZT.age.max
## fertility
## total fecundity from Allainé et al. 1987 (Oecologia)
## lnF = 2.719 - 0.211lnM (all mammals)
ZT.F.pred1 <- exp(2.719 - (0.211*log(ZT.mass*1000)))/2 # divided by 2 for females
ZT.F.pred <- ZT.F.pred1 * (365/as.numeric((mean(VOMBAT.IBI) + (VU.IBI - VU.IBI.pred)) + (MACROPOD.F.corr.b * log10(ZT.mass))))
## age at primiparity
## lnalpha = 0.214 + 0.263*lnM (https://dx.doi.org/10.1093%2Fgerona%2F62.2.149)
ZT.alpha1 <- ceiling(exp(-1.34 + (0.214*log(ZT.mass*1000))))
ZT.alpha <- ZT.alpha1
## define m function with age
ZT.m.vec <- c(rep(0, ZT.alpha-1), rep(0.75*ZT.F.pred, round(ZT.alpha/2,0)), rep(ZT.F.pred, (ZT.age.max+1-((ZT.alpha-1+round(ZT.alpha/2,0))))))
ZT.m.sd.vec <- 0.05*ZT.m.vec
plot(ZT.age.vec, ZT.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
# fit sigmoidal function
# logistic power function y = a / (1+(x/b)^c)
ZT.m.dat <- data.frame(ZT.age.vec, ZT.m.vec)
param.init <- c(0.1, 2, -5)
ZT.fit.logp <- nls(ZT.m.vec ~ a / (1+(ZT.age.vec/b)^c),
data = ZT.m.dat,
algorithm = "port",
start = c(a = param.init[1], b = param.init[2], c = param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
ZT.fit.logp.summ <- summary(ZT.fit.logp)
plot(ZT.age.vec, ZT.m.vec, type="b", pch=19, xlab="age (yrs)", ylab="m")
ZT.age.vec.cont <- seq(0,max(ZT.age.vec),1)
ZT.pred.p.m <- coef(ZT.fit.logp)[1] / (1+(ZT.age.vec.cont/coef(ZT.fit.logp)[2])^coef(ZT.fit.logp)[3])
ZT.pred.p.mm <- ifelse(ZT.pred.p.m > 1, 1, ZT.pred.p.m)
lines(ZT.age.vec.cont, ZT.pred.p.mm,lty=2,lwd=3,col="red")
## survival
## mean adult survival (McCarthy et al. 2008 Am Nat)
## ln{-ln[s(t)]} = ln(a) + bln(M) + ln (t)
ln.a.s <- -0.5; b.s <- -0.25
ZT.s.tran <- ln.a.s + b.s*log(ZT.mass*1000) + log(1)
ZT.s.ad.yr <- exp(-exp(ZT.s.tran))
# Siler hazard h(x) (Gurven et al. 2007)
a1 <- 1 - (1.01*ZT.s.ad.yr) # initial infant mortality rate (also known as αt)
b1 <- 2.7 # rate of mortality decline (also known as bt)
a2 <- 1 - ZT.s.ad.yr # age-independent mortality (exogenous mortality due to environment); also known as ct
a3 <- 0.1e-04 # initial adult mortality rate (also known as βt)
b3 <- 0.02 # rate of mortality increase
longev <- ZT.age.max
x <- seq(0,longev,1) # age vector
h.x <- a1 * exp(-b1*x) + a2 + a3 * exp(b3 * x) # Siler's hazard model
plot(x,h.x,pch=19,type="l")
plot(x,log(h.x),pch=19,type="l")
l.x <- exp((-a1/b1) * (1 - exp(-b1*x))) * exp(-a2 * x) * exp(a3/b3 * (1 - exp(b3 * x))) # Siler's survival (proportion surviving) model
init.pop <- 10000
lx <- round(init.pop*l.x,0)
len.lx <- length(lx)
dx <- lx[1:(len.lx-1)]-lx[2:len.lx]
qx <- dx/lx[1:(length(lx)-1)]
ZT.Sx <- c(0.995*ZT.s.ad.yr, 1 - qx)
plot(x, ZT.Sx, pch=19, type="l", xlab="age (years)", ylab="Sx")
ZT.s.sd.vec <- 0.05*ZT.Sx
## create matrix
ZT.popmat <- matrix(data = 0, nrow=ZT.age.max+1, ncol=ZT.age.max+1)
diag(ZT.popmat[2:(ZT.age.max+1),]) <- ZT.Sx[-(ZT.age.max+1)]
ZT.popmat[ZT.age.max+1,ZT.age.max+1] <- ZT.Sx[ZT.age.max+1]
ZT.popmat[1,] <- ZT.pred.p.mm
colnames(ZT.popmat) <- c(0:ZT.age.max)
rownames(ZT.popmat) <- c(0:ZT.age.max)
ZT.popmat.orig <- ZT.popmat ## save original matrix
## matrix properties
max.lambda(ZT.popmat.orig) ## 1-yr lambda
ZT.lm.pred
max.r(ZT.popmat.orig) # rate of population change, 1-yr
ZT.ssd <- stable.stage.dist(ZT.popmat.orig) ## stable stage distribution
plot(ZT.age.vec, ZT.ssd, type="l", pch=19, xlab="age (yrs)", ylab="ssd")
R.val(ZT.popmat.orig, ZT.age.max) # reproductive value
ZT.gen.l <- G.val(ZT.popmat.orig, ZT.age.max) # mean generation length
## initial population vector
area <- 500*500 # km × km
ZT.pop.found <- round(area*ZT.D.pred, 0) # founding population size (estimated density * 100 × 100 km region [10,000 km2])
ZT.init.vec <- ZT.ssd * ZT.pop.found
#################
## project
## set time limit for projection in 1-yr increments
yr.st <- 1
#************************
yr.end <- round(40*ZT.gen.l, 0) # set projection end date
#************************
t <- (yr.end - yr.st)
ZT.tot.F <- sum(ZT.popmat.orig[1,])
ZT.popmat <- ZT.popmat.orig
yr.vec <- seq(yr.st,yr.end)
## set population storage matrices
ZT.n.mat <- matrix(0, nrow=ZT.age.max+1,ncol=(t+1))
ZT.n.mat[,1] <- ZT.init.vec
## set up projection loop
for (i in 1:t) {
ZT.n.mat[,i+1] <- ZT.popmat %*% ZT.n.mat[,i]
}
ZT.n.pred <- colSums(ZT.n.mat)
yrs <- seq(yr.st, yr.end, 1)
plot(yrs, log10(ZT.n.pred),type="l",lty=2,pch=19,xlab="year",ylab="log10 N")
# compensatory density feedback
ZT.K.max <- 1*ZT.pop.found
ZT.K.vec <- c(1, ZT.K.max/2, 0.75*ZT.K.max, ZT.K.max)
ZT.red.vec <- c(1,0.975,0.945,0.9018)
plot(ZT.K.vec, ZT.red.vec,pch=19,type="b")
ZT.Kred.dat <- data.frame(ZT.K.vec, ZT.red.vec)
# logistic power function a/(1+(x/b)^c)
ZT.param.init <- c(1, 2*ZT.K.max, 2)
ZT.fit.lp <- nls(ZT.red.vec ~ a/(1+(ZT.K.vec/b)^c),
data = ZT.Kred.dat,
algorithm = "port",
start = c(a = ZT.param.init[1], b = ZT.param.init[2], c = ZT.param.init[3]),
trace = TRUE,
nls.control(maxiter = 1000, tol = 1e-05, minFactor = 1/1024))
ZT.fit.lp.summ <- summary(ZT.fit.lp)
plot(ZT.K.vec, ZT.red.vec, pch=19,xlab="N",ylab="reduction factor")
ZT.K.vec.cont <- seq(1,2*ZT.pop.found,1)
ZT.pred.lp.fx <- coef(ZT.fit.lp)[1]/(1+(ZT.K.vec.cont/coef(ZT.fit.lp)[2])^coef(ZT.fit.lp)[3])
lines(ZT.K.vec.cont, ZT.pred.lp.fx, lty=3,lwd=3,col="red")
ZT.a.lp <- coef(ZT.fit.lp)[1]
ZT.b.lp <- coef(ZT.fit.lp)[2]
ZT.c.lp <- coef(ZT.fit.lp)[3]
## compensatory density-feedback deterministic model
## set population storage matrices
ZT.n.mat <- matrix(0, nrow=ZT.age.max+1, ncol=(t+1))
ZT.n.mat[,1] <- ZT.init.vec
ZT.popmat <- ZT.popmat.orig
## set up projection loop
for (i in 1:t) {
ZT.totN.i <- sum(ZT.n.mat[,i])
ZT.pred.red <- as.numeric(ZT.a.lp/(1+(ZT.totN.i/ZT.b.lp)^ZT.c.lp))
diag(ZT.popmat[2:(ZT.age.max+1),]) <- (ZT.Sx[-(ZT.age.max+1)])*ZT.pred.red
ZT.popmat[ZT.age.max+1,ZT.age.max+1] <- (ZT.Sx[ZT.age.max+1])*ZT.pred.red
ZT.popmat[1,] <- ZT.pred.p.mm
ZT.n.mat[,i+1] <- ZT.popmat %*% ZT.n.mat[,i]
}
ZT.n.pred <- colSums(ZT.n.mat)
plot(yrs, ZT.n.pred, type="l",lty=2,pch=19,xlab="year",ylab="N")
abline(h=ZT.pop.found, lty=2, col="red", lwd=2)
## stochatic projection with density feedback
## set storage matrices & vectors
iter <- 100
itdiv <- iter/10
ZT.n.sums.mat <- matrix(data=NA, nrow=iter, ncol=(t+1))
ZT.s.arr <- ZT.m.arr <- array(data=NA, dim=c(t+1, ZT.age.max+1, iter))
for (e in 1:iter) {
ZT.popmat <- ZT.popmat.orig
ZT.n.mat <- matrix(0, nrow=ZT.age.max+1,ncol=(t+1))
ZT.n.mat[,1] <- ZT.init.vec
for (i in 1:t) {
# stochastic survival values
ZT.s.alpha <- estBetaParams(ZT.Sx, ZT.s.sd.vec^2)$alpha
ZT.s.beta <- estBetaParams(ZT.Sx, ZT.s.sd.vec^2)$beta
ZT.s.stoch <- rbeta(length(ZT.s.alpha), ZT.s.alpha, ZT.s.beta)
# stochastic fertilty sampler (gaussian)
ZT.fert.stch <- rnorm(length(ZT.popmat[,1]), ZT.pred.p.mm, ZT.m.sd.vec)
ZT.m.arr[i,,e] <- ifelse(ZT.fert.stch < 0, 0, ZT.fert.stch)
ZT.totN.i <- sum(ZT.n.mat[,i], na.rm=T)
ZT.pred.red <- ZT.a.lp/(1+(ZT.totN.i/ZT.b.lp)^ZT.c.lp)
diag(ZT.popmat[2:(ZT.age.max+1),]) <- (ZT.s.stoch[-(ZT.age.max+1)])*ZT.pred.red
ZT.popmat[ZT.age.max+1,ZT.age.max+1] <- (ZT.s.stoch[ZT.age.max+1])*ZT.pred.red
ZT.popmat[1,] <- ZT.m.arr[i,,e]
ZT.n.mat[,i+1] <- ZT.popmat %*% ZT.n.mat[,i]
ZT.s.arr[i,,e] <- ZT.s.stoch * ZT.pred.red
} # end i loop
ZT.n.sums.mat[e,] <- ((as.vector(colSums(ZT.n.mat))/ZT.pop.found))
if (e %% itdiv==0) print(e)
} # end e loop
ZT.n.md <- apply(ZT.n.sums.mat, MARGIN=2, median, na.rm=T) # mean over all iterations
ZT.n.up <- apply(ZT.n.sums.mat, MARGIN=2, quantile, probs=0.975, na.rm=T) # upper over all iterations
ZT.n.lo <- apply(ZT.n.sums.mat, MARGIN=2, quantile, probs=0.025, na.rm=T) # lower over all iterations
par(mfrow=c(1,3))
plot(yrs,ZT.n.md,type="l", main = "", xlab="year", ylab="pN1", lwd=2, ylim=c(0.95*min(ZT.n.lo),1.05*max(ZT.n.up)))
lines(yrs,ZT.n.lo,lty=2,col="red",lwd=1.5)
lines(yrs,ZT.n.up,lty=2,col="red",lwd=1.5)
ZT.s.add <- ZT.m.add <- rep(0, ZT.age.max+1)
for (m in 1:iter) {
ZT.s.add <- rbind(ZT.s.add, ZT.s.arr[ceiling(ZT.gen.l):(t+1),,m])
ZT.m.add <- rbind(ZT.m.add, ZT.m.arr[ceiling(ZT.gen.l):(t+1),,m])