rm(list=ls())

N0 <- 200
mu <- 0.001672504
sigma <-  0.236635 
N <- array(0,c(100,25))
N[,1] <- rep(N0,100)
for (i in 1:100){
for (j in 1:24){
N[i,j+1] <-exp(rnorm(1,mu,sigma))*N[i,j]
}
}
t <- seq(1,25,1)
plot(t,N[1,], type="n", ylim=c(0,2000),ylab="N")
for (k in 1:100){
lines(t,N[k,], col=runif(1,1,10))
}
 


#######################################
lag <- 100
x <- array(0,c(1,lag))
r = 3.80000000001
x[1] <- 0.1 #####

for (i in 2:lag) {
x[i] <- r*x[i-1]*(1-x[i-1])
}

par( mfrow=c(2,1))
t <- seq(1,lag,1)
plot(t,x, type="n",ylab="N")
lines(t,x)
a <- x[1:(length(x)-1)]
b <- x[2:length(x)]
plot(a,b, type="n",xlab="N", ylab="Nt+1")
points(a,b,pch=16)
par( mfrow=c(1,1))
x[99]


#################################

K <- 100
ri <- 5
theta <- c(3,1,.3)
xi <- seq(1,100,1)
Xi <- array(0,c(100,3))

for (i in 1:3){
Xi[,i] <- xi*exp(ri*(1-(xi/K)^theta[i]))}

plot(xi,log(Xi[,1]/xi), type ="n",xlab="Nt",ylab="log(Nt/Nt+1)")
lines(xi,log(Xi[,1]/xi),col="red")
lines(xi,log(Xi[,2]/xi),col="blue")
lines(xi,log(Xi[,3]/xi),col="yellow")

#################################
rr <- seq(2.9,4,0.001)
ini <- seq(0.3,1,0.01)
m <- array(0,c(length(ini),length(rr)))

for (j in 1:length(rr)) {
for (k in 1:(length(ini-1))) {
		x[1] <- ini[k]
		for (i in 2:300) 	{
		x[i] <- rr[j]*x[i-1]*(1-x[i-1])
}
		m[k,j]<- x[299]
}
}

###################################



plot(rr,m[length(ini),], type ="n",ylim=c(0,1))
for (i in 1:(length(ini)-1)){
points(rr,m[i,1:length(rr)],pch=16, cex=0.00001, col=round(runif(1)*10,0))
}