Population dynamics - part 4

The Logistic Map

The logistic map is the discrete case of the logistic equation, given by:   $\frac {\mathrm{d}y}{\mathrm{d}t}=ry(1-\frac{y}{Y})$

We then approximate to deduce the discrete case:$$\begin{array}{rl}\frac{y_{n+1}-y_n}{\mathrm{\Delta }t}& \approx r{y}_n(1-\frac{y_n}{Y})\\ y_{n+1}& \approx r\mathrm{\Delta }t{y}_n(1-\frac{y_n}{Y})+y_n\\ y_{n+1}& =(1+r\mathrm{\Delta }t)y_n-\frac{r\mathrm{\Delta }t}{Y}{(y_n)}^2\\ y_{n+1}& =(1+r\mathrm{\Delta }t)y_n(1-(\frac{r\mathrm{\Delta }t}{1+r\mathrm{\Delta }t})\frac{y_n}{Y})\end{array}$$

Let $\lambda=1+r\Delta t$ and $x_n=\frac {r\Delta t}{1+r \Delta t} \frac {y_n}{Y}$ . Then our equation becomes: $$x_{n+1}=\lambda x_n(1-x_n)$$ This is the logistic map. We can also think of it as a function $x_{n+1}=f(x_n)$.

Finding Equilibrium Points

Question:   A fixed point implies $x_{n+1}=x_n$ . Find the fixed points by solving $$\lambda x_n(1-x_n)=x_n$$ To determine the stability of these points, we are going to find the stability, by investigating the function for values nearby the equilibrium points.

Start by supposing that $x_n=X$ is a fixed point. This means that $f(X)=X$. 

To find a value near the equilibrium point, let $x_n=X+\epsilon{_n}$ where $\epsilon_n < < 1$. Then using the Taylor expansion: $$\begin{array}{rl}{x}_{n+1}& =f(x_n)\\ X+{ϵ}_{n+1}& =f(X+{ϵ}_n)\\ & =f(X)+{ϵ}_n{f}^{\prime }(X)+...\end{array}$$

We neglect the higher-order terms to get: $$X+{ϵ}_{n+1}=f(X)+{ϵ}_n{f}^{\prime }(X)$$ Now from above we saw that $f(X)=X$ , so we can simplify to get: $${ϵ}_{n+1}\approx f^{\prime }(X){ϵ}_n$$ A fixed point, X, is then stable if:   $\Bigg|\frac{\epsilon_{n+1}}{\epsilon_n}\Bigg | =\Bigg |f'(X)\Bigg | < 1$

Question:  Given that $f'(x)=\lambda-2\lambda x$ , find the stability of the fixed points $x_n=0$  and  $x_n=1-\frac{1}{\lambda}$

Different Cases of Stability

Below are some graphs of the logistic map for different values of $\lambda$ .

         Case 1: $\lambda< 1$

Only fixed point is 0, which is stable:

Image

         Case 2: $1< \lambda < 2$

Unstable fixed point at 0 and stable fixed point at $1-\frac{1}{\lambda}$

Image

Question:  Can you find the stability for the case $2< \lambda < 3$  ? 

Below is a picture of some fantastic fractal behaviour which occurs for $3< \lambda< 4$.

Image

Question:  Can you relate these values of $\lambda$ to what would actually be occuring in a population of organisms?