## MA34810 - Mathematical Models of Biological Systems

#### Problem Sheet 2 (Period Doubling leads to Chaos)

The discrete logistic map can be used to describe the seasonal changes of a bounded population.

The tools below allow you to explore the dynamics of the map using the computer.

#### Demonstration 1: the cobweb diagram

Open the diagram here.

On the left hand side (the Time series) you see the value of u_{t} plotted at each time step. There are 300 time steps.

On the right hand side (the Cobweb plot) you see the value of u at time t+1 plotted against its value at the previous time step.

Move the sliders to vary the value of r and the initial point u_{0}. Click on the "+" to give each slider a precise value.

#### Demonstration 2: the bifurcation diagram

The bifurcation diagram summarizes the cobweb diagram by plotting the stable steady states of u_{t} against the parameter r.

Open the bifurcation diagram here.

This version lets you explore and enlarge different areas of the map to show its fractal nature. As the magnification increases, it is helpful to increase the number of points that are plotted.