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Appendix: Details concerning simulations including annotated R code.
Summary:
Simulations are all based on a simple process model in which colonization is solely dependent on the
number of occupied cells it touches (including diagonal neighbors) and extinction depends only on
habitat quality. In all instances, the landscape wraps in all directions. In other words, a pixel in the first
column includes pixels in the last columns as its neighbors and vice-versa. We conducted 100
simulations across four landscapes under four scenarios: 1) high colonization (0.8 probability of
colonization if all neighbors are occupied and decreasing by 0.1 for each unoccupied neighbor) at
equilibrium, 2) low colonization (0.4 probability of colonization if all neighbors are occupied) at
equilibrium, 3) high colonization at disequilibrium, and 4) low colonization at disequilibrium. In all cases,
extinction was 0.1 in low quality habitat and 0.8 in high quality. We also choose one set of simulations
and plotted occupancy as a function of the number of good neighbors (landscape covariate) or occupied
patches (autologistic) for figure 4
Annotated R code:
#Simulations to create figure 3
# define update function which updates occupancies one time step based on input occupancy, p,
# as well as extinction rate, E, and colonization rate, G, when all neighbors are occupied. Note that there
# are no edges in these simulations…patches on one side are neighbors to patches on the opposite side.
update<-function(p,E,G){
d<-dim(p)[1]
neigh<-p[c(2:d,1),]+p[c(d,1:(d-1)),]+
p[,c(2:d,1)]+p[,c(d,1:(d-1))]+
p[c(2:d,1),c(2:d,1)]+
p[c(d,1:(d-1)),c(d,1:(d-1))]+
p[c(d,1:(d-1)),c(2:d,1)]+
p[c(2:d,1),c(d,1:(d-1))]
out<-matrix(rbinom((d*d),1,p*(1-E)+(1-p)*(G*neigh/8)),nrow=d,ncol=d)
return(out)}
#define loop function which steps forward a number of time steps, Nloop, based on initial occupancy,
# init.psi, and extinction, e, and colonization, g, as in update function.
loop<-function(Nloop,init.psi,e,g=0.6){
occ<-init.psi
for (j in 1:Nloop){occ<-update(occ,E=e,G=g)}
return(occ)}
#set random number seed
set.seed(1)
#create arrays to save simulated equilibrium occupancies and associated habitat quality for equilibrium
# (eq) and disequilibrium (diseq) simulations. First dimension is based on the number of patches in each
# simulations, second is based on the number of simulations, third dimension is based on the four
# different landscapes, and in the case of simsave arrays fourth dimension is for low and high
# colonization rates.
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simsave.eq<-array(NA,dim=c(1600,100,4,2))
hqsave<-array(NA,dim=c(1600,100,4))
simsave.diseq<-array(NA,dim=c(1600,100,4,2))
#Simulate! See annotations within loop.
for (k in 1:100){
## First Landscape - 20% Random- Equilibrium simulations
# create matrix, hq, with worse, 0, and better, 1, habitat
hq<-matrix(sample(c(rep(0,1280),rep(1,320)),1600,replace=FALSE),nrow=40,ncol=40)
#convert hq to extinction rate
eps<-.8-.7*hq
#guess at equilibrium rate assuming colonization of 50% of maximum
eqO<-.2/(.2+eps)
#simulate initial occupancy
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
#simulate 100 time steps to reach dynamic equilibrium (more than enough)
temp<-loop(100,IO,eps,.4)
#save simulations and habitat info
simsave.eq[,k,1,1]<-as.vector(temp)
hqsave [,k,1]<-as.vector(hq)
#repeat with higher colonization rate and save
eqO<-.4/(.4+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.8)
simsave.eq[,k,1,2]<-as.vector(temp)
## First Landscape - 20% Random- Disequilibrium simulations (only difference is 12 time steps
# instead of 100 and initiall occupancy is chosen randomly to cover 2.5% of patches)
IO<-matrix(sample(c(rep(0,1560),rep(1,40)),1600,replace=FALSE),nrow=40,ncol=40)
temp<-loop(12,IO,eps,.4)
simsave.diseq[,k,1,1]<-as.vector(temp)
temp<-loop(12,IO,eps,.8)
simsave.diseq[,k,1,2]<-as.vector(temp)
## Second Landscape - 50% Random- Equilibrium simulations
hq<-matrix(sample(c(rep(0,800),rep(1,800)),1600,replace=FALSE),nrow=40,ncol=40)
eps<-.8-.7*hq
eqO<-.2/(.2+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.4)
simsave.eq[,k,2,1]<-as.vector(temp)
hqsave [,k,2]<-as.vector(hq)
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eqO<-.4/(.4+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.8)
simsave.eq[,k,2,2]<-as.vector(temp)
## Second Landscape - 50% Random- Disequilibrium simulations
IO<-matrix(sample(c(rep(0,1560),rep(1,40)),1600,replace=FALSE),nrow=40,ncol=40)
temp<-loop(12,IO,eps,.4)
simsave.diseq[,k,2,1]<-as.vector(temp)
temp<-loop(12,IO,eps,.8)
simsave.diseq[,k,2,2]<-as.vector(temp)
## Third Landscape - 80% Random- Equilibrium simulations
hq<-matrix(sample(c(rep(0,320),rep(1,1280)),1600,replace=FALSE),nrow=40,ncol=40)
eps<-.8-.7*hq
eqO<-.2/(.2+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.4)
simsave.eq[,k,3,1]<-as.vector(temp)
hqsave [,k,3]<-as.vector(hq)
eqO<-.4/(.4+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.8)
simsave.eq[,k,3,2]<-as.vector(temp)
## Third Landscape - 80% Random- Disequilibrium simulations
IO<-matrix(sample(c(rep(0,1560),rep(1,40)),1600,replace=FALSE),nrow=40,ncol=40)
temp<-loop(12,IO,eps,.4)
simsave.diseq[,k,3,1]<-as.vector(temp)
temp<-loop(12,IO,eps,.8)
simsave.diseq[,k,3,2]<-as.vector(temp)
## Fourth Landscape – 50% Block- Equilibrium simulations
hq<-matrix(c(rep(1,800),rep(0,800)),nrow=40,ncol=40)
eps<-.8-.7*hq
eqO<-.2/(.2+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.4)
simsave.eq[,k,4,1]<-as.vector(temp)
hqsave [,k,4]<-as.vector(hq)
eqO<-.4/(.4+eps)
IO<-matrix(rbinom(1600,1,eqO),nrow=40,ncol=40)
temp<-loop(100,IO,eps,.8)
simsave.eq[,k,4,2]<-as.vector(temp)
## Fourth Landscape – 50% Block- Disequilibrium simulations
IO<-matrix(sample(c(rep(0,1560),rep(1,40)),1600,replace=FALSE),nrow=40,ncol=40)
temp<-loop(12,IO,eps,.4)
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simsave.diseq[,k,4,1]<-as.vector(temp)
temp<-loop(12,IO,eps,.8)
simsave.diseq[,k,4,2]<-as.vector(temp)
}
#summarize results for each landscape and simulation by habitat quality occupancy
occbyhab.eq<-array(NA,dim=c(2,100,4,2))
occbyhab.diseq<-array(NA,dim=c(2,100,4,2))
for (i in 1:100){
for (j in 1:4){
occbyhab.eq[1,i,j,1]<-mean(subset(simsave.eq[,i,j,1],hqsave[,i,j]==0))
occbyhab.eq[2,i,j,1]<-mean(subset(simsave.eq[,i,j,1],hqsave[,i,j]==1))
occbyhab.eq[1,i,j,2]<-mean(subset(simsave.eq[,i,j,2],hqsave[,i,j]==0))
occbyhab.eq[2,i,j,2]<-mean(subset(simsave.eq[,i,j,2],hqsave[,i,j]==1))
occbyhab.diseq[1,i,j,1]<-mean(subset(simsave.diseq[,i,j,1],hqsave[,i,j]==0))
occbyhab.diseq[2,i,j,1]<-mean(subset(simsave.diseq[,i,j,1],hqsave[,i,j]==1))
occbyhab.diseq[1,i,j,2]<-mean(subset(simsave.diseq[,i,j,2],hqsave[,i,j]==0))
occbyhab.diseq[2,i,j,2]<-mean(subset(simsave.diseq[,i,j,2],hqsave[,i,j]==1))
}}
#For figure 4 analysis summarize
habitat.1<-function(H){
(H[c(2:40,1),]+H[c(40,1:(40-1)),]+H[,c(2:40,1)]+H[,c(40,1:(40-1))]+H[c(2:40,1),c(2:40,1)]+
H[c(40,1:(40-1)),c(40,1:(40-1))]+H[c(40,1:(40-1)),c(2:40,1)]+H[c(2:40,1),c(40,1:(40-1))])/8}
nonpar<-array(NA,dim=c(18,100,4,4))
par(mfrow=c(1,2))
for (k in 1:100){
for (j in 1:4){
t1<-simsave.eq[,k,j,1]
t2<-hqsave[,k,j]
t3<-as.vector(habitat.1(matrix(hqsave[,k,j],nrow=40)))
t4<-as.vector(habitat.1(matrix(simsave.eq[,k,j,1],nrow=40)))
for (i in 1:9){
t5<-subset(t1,t2==0&t3==((i-1)/8))
nonpar[i,k,j,1]<-ifelse(length(t5)>9,mean(t5),NA)
t5<-subset(t1,t2==1&t3==((i-1)/8))
nonpar[(i+9),k,j,1]<-ifelse(length(t5)>9,mean(t5),NA)
t5<-subset(t1,t2==0&t4==((i-1)/8))
nonpar[i,k,j,2]<-ifelse(length(t5)>9,mean(t5),NA)
t5<-subset(t1,t2==1&t4==((i-1)/8))
nonpar[(i+9),k,j,2]<-ifelse(length(t5)>9,mean(t5),NA)
}
}}
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