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# The predator-prey model revisited

Consider the two-dimensional example:

This is a predator-prey model, with xn the prey and yn the predators. We have seen in class that if the parameter b is large enough, then there is a coexistent state in which there are both predators and prey. However, we also showed that if b gets too large, this coexistent state is unstable. So what happens? The loss of stability is through a pair of complex eigenvalues, so theory says we can expect an invariant circle (almost periodic behavior). Here is the ODE file:
# discrete predator prey
x'=r*x*(1-x)-a*x*y
y'=g*y+b*x*y
par g=.8,b=1.5,a=1,r=2.5
init y=1.17,x=.13
@ meth=discrete,total=200
done

We will look at the phaseplane. Click on /ppcViewaxes /ppc2D ( ) and fill in the resulting dialog box as follows:
and click on . You have now set it up to plot prey along the X-axis and predators along the Y-axis. Lastly, we want to make sure only dots are plotted so we will edit the graphics curve. Click on ( ) and accept 0 as the curve to edit by clicking . Now change the Line type in the dialog box from 1 to 0 and click on . Linetype 0 is just a point. Run the simulation ( ). You should just see a dot. Now change the initial conditions to (x=.6, y=.01) to simulate a whole lot of rabbits and a small number of foxes. Click on in the Initial Data Windowand watch those rabbits get eaten. Not the spiraling in toward the equilibrium. Change the parameter b to 2.2, erase the screen (in Main Window, tap ), and rerun the simulation. You should see an invariant circle appear in which neither settles to a steady state, but rather, they tend to a closed curve. Add more iterates to the simulation by clicking and typing in, say 800. The circle is more evident. Quit if you want, or continue to fool around. To quit tap .

Next: Homework - optional Up: No Title Previous: Cobwebbing
G. Bard Ermentrout
1999-09-17