01 - Discrete Models

Discrete-Time Model

Let $N_t$ denote the number of individuals in timestep $t$ then

$$N_{t+1}=f(N_t)$$

where $f$ is some function

Note that $f(N_t)$ is usually in form $f(N_t)=N_{t}g(N_t)$

Exponential Model

Models population where

1. Average number of births for each individual by $b$

2. Probability of Individual dying is $d$

Let

$$N_{t+1}=(1+b-d)N_t=r{N}_t$$

where $r>0$

Then the solution is

$$N_t=r^t{N}_0=\{\begin{array}{ll}\infty & \text{ if }r>1\\ N_0& \text{ if }r=1\text{ as }t\to \infty \\ 0& \text{ if }r<1\end{array}$$

Logistic Model

Models Population with a Carrying Capacity so population size won’t exceed a limit

Let $K$ be the carrying capacity of the model then

$$N_{t+1}=N_t+r{N}_t\left(1-\frac{N_t}{K}\right)$$

where $r>0$ and $K>0$

Note that for large $N_t$ then the next timestep can be negative

Ricker Model

Models Population with carrying capacity $K$ but with non-negative population

$$N_{t+1}=N_t\exp \left[r\left(1-\frac{N_t}{k}\right)\right]$$

where $r>0$ and $K>0$

Hassel Model

Models temporal evolution of population

$$N_{t+1}=\frac{r{N}_t}{(1+a{N}_t{)}^b}$$

where $r>0$, $a>0$ and $b>0$

Cobwebbing

Let

$$N_{t+1}=f(N_t)$$

Cobweb Map is a graph of $N_{t+1}$ as a function of $N_t$

Plot

$$N_{t+1}=f(N_t)\text{ and }{N}_{t+1}=N_t$$

Then

1. Start on $x$-axis at initial value $N_0$

2. Plot vertically to line $N_{t+1}=f(N_t)$

3. Plot horizontally across to line $N_{t+1}=N_t$

4. Repeat steps $2$-$3$ indefinitely