DynamicModelsPrimer.Rmd

---
title: "Dynamic Models Primer"
output: html_notebook
---

## Random walk
```{r}
NT = 30
x = rep(NA,NT)

x[1] = 5
for(t in 1:(NT-1)){
  x[t+1] = x[t]
}
plot(x,type="b")
```

## rho*X
```{r}
NT = 30
x = rep(NA,NT)

rho = 0.9
x[1] = 5
for(t in 1:(NT-1)){
  x[t+1] = rho*x[t]
}
plot(x,type='b',ylim=c(0,x[1]))
abline(h=0,col="orange")
```

## b0 + rho*X
```{r}
NT = 30
x = rep(NA,NT)

b0 = 1
rho = 0.9
x[1] = 5
for(t in 1:(NT-1)){
  x[t+1] = b0 + rho*x[t]
}
equlib = b0/(1-rho)
plot(x,type='b',ylim=c(x[1],equlib))
abline(h=equlib,col="orange")
```

## rho*X + B2*X^2
```{r}
NT = 30
x = rep(NA,NT)

r = 0.5
K = 10
rho = 1+r
b2 = -r/K
x[1] = 5
for(t in 1:(NT-1)){
  x[t+1] = rho*x[t] + b2*x[t]^2
}
plot(x,type='b',ylim=c(x[1],K))
abline(h=K,col="orange")
```
This model is equivalent to logistic growth

## X + B1*Z
```{r}
NT = 30
x = rep(NA,NT)
Z = sin(1/12*(2*pi)*(0:(NT-1)))
#plot(Z,type="b")

b1 = 0.5
x[1] = 0
for(t in 1:(NT-1)){
  x[t+1] = x[t] + b1*Z[t]
}
#equlib = b2/rho # = K
plot(x,type='b',ylim=range(c(x,Z)))
lines(Z,type='l',col="blue")
abline(h=0,col="orange")
abline(v=seq(1,30,by=6),col="orange")
```

## rho*X + B1*Z
```{r}
NT = 30
x = rep(NA,NT)
Z = sin(1/12*(2*pi)*(0:(NT-1)))
plot(Z,type="b")

rho = 0.9
b1 = 0.5
x[1] = 5
for(t in 1:(NT-1)){
  x[t+1] = rho*x[t] + b1*Z[t]
}
#equlib = b2/rho # = K
plot(x,type='b')#,ylim=c(x[1],equlib))
#abline(h=equlib,col="orange")
```

## K = a0 + a1*Z
```{r}
NT = 30
Z = sin(1/12*(2*pi)*(0:(NT-1)))
plot(Z,type="b")

rho = c(0.25,0.9,1.9)
x = K = matrix(NA,length(rho),NT)
a0 = 5
a1 = 3
for(r in seq_along(rho)){
  x[r,1] = 5
  for(t in 1:(NT-1)){
    K[r,t] = a0 + a1*Z[t+1]
    x[r,t+1] = x[r,t] + rho[r]*x[r,t]*(1-x[r,t]/K[r,t])
  }
}
plot(x[1,],type='l',ylim=range(c(x,Z)))
lines(x[2,],col=2)
lines(x[3,],col=3)
lines(Z,col="blue")
#lines(K[1,],col="purple")
abline(h=0,col="orange")
abline(v=seq(1,30,by=6),col="orange")
legend("topright",legend=c(rho,"Z"),lwd=2,col=c(1:3,"blue"))

for(i in seq_along(rho)){
  plot(Z,x[i,],col=i,asp=1)
  abline(lm(x[i,]~Z),lty=2)
}
```

## delta Z
```{r}
NT = 30
Z = sin(1/12*(2*pi)*(0:(NT-1)))

rho = c(0.25,0.6,0.95)
x = matrix(NA,length(rho),NT)
b1 = 0.5
for(r in seq_along(rho)){
  x[r,1] = 0
  for(t in 1:(NT-1)){
    x[r,t+1] = rho[r]*x[r,t] + b1*(Z[t+1]-Z[t])
  }
}
plot(x[1,],type='l',ylim=range(c(x,Z)))
lines(x[2,],col=2)
lines(x[3,],col=3)
lines(Z,col="blue")
#lines(K[1,],col="purple")
abline(h=0,col="orange")
abline(v=seq(1,30,by=6),col="orange")
legend("topright",legend=c(rho,"Z"),lwd=2,col=c(1:3,"blue"))

for(i in seq_along(rho)){
  plot(Z,x[i,],col=i,asp=1)
  abline(lm(x[i,]~Z),lty=2)
}
```

## b0 + rho*X + rho2*X[t-1]  AR(2)
```{r}
NT = 30
b0 = 1
rho = 0.7
rho2 = c(0.2,-0.2)
x = matrix(NA,length(rho2),NT)
for(r in seq_along(rho2)){
  x[r,1] = 5
  for(t in 1:(NT-1)){
    if(t>1){lag = rho2[r]*x[r,t-1]} else {lag=0}
    x[r,t+1] = b0 + rho*x[r,t] + lag
  }
}
plot(x[1,],type='l',ylim=range(x))
lines(x[2,],col=2)
legend("topright",legend=c(rho2),lwd=2,col=c(1:2))
```