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gibbscap1=function(nsimu,n1,c2,c3,N0=n1/runif(1),r10,r20){
# GIBBS SAMPLING FOR THE 2-STAGE OPEN POPULATION MODEL
N=rep(0,nsimu)
p=rep(0,nsimu)
q=rep(0,nsimu)
r1=rep(0,nsimu)
r2=rep(0,nsimu)
N[1]=N0
r1[1]=r10
r2[1]=r20
nplus=n1+c2+c3
for (i in 2:nsimu){
uplus=N[i-1]-r1[i-1]-c2+n1-r1[i-1]-r2[i-1]-c3
p[i]=rbeta(1,nplus+1,uplus+1)
q[i]=rbeta(1,r1[i-1]+r2[i-1]+1,2*n1-2*r1[i-1]-r2[i-1]+1)
N[i]=n1+rnbinom(1,n1,p[i])
mm=min(n1-r2[i-1]-c3,n1-c2)
pq=q[i]/(1+(1-q[i])*(1-p[i]))
pr=lchoose(n1-c2,0:mm)+(0:mm)*log(pq)+lchoose(n1-(0:mm),r2[i-1]+c3)
r1[i]=sample(0:mm,1,prob=exp(pr-max(pr)))
r2[i]=rbinom(1,n1-r1[i]-c3,q[i]/(1+(1-q[i])*(1-p[i])))
}
list(N=N,p=p,q=q,r1=r1,r2=r2)
}
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