## Nonlinear Predator Prey Model

This model has its roots in a Computer Recreations column in Scientific American in December 1984. Each site has three states: 0 = vacant, 1 = prey (fish), 2 = predators (sharks), and eight neighbors: the sites with each coordinate differing by at most one.

Fish are born at vacant sites at rate b times the fraction of neighbors occupied by fish.

Each shark at rate 1 inspects q neighboring sites. It eats the first fish it finds and moves there.

A shark that has just eaten gives birth with probability b. A shark that finds no fish dies with probability delta.

Finally there is stirring at rate nu: for each pair of neighboring sites x and y we exchange the values at x and at y at rate nu.

The mean field ODE is: Since fish are a contact process with no death, there is a boundary equilibrium at (1,0). However, the second equation shows that (1,0) is always a saddle point. Further, starting with the second equation and solving, we see that the ODE has a unique interior fixed point. Linearizing around the fixed point shows that it is locally attracting when q < 3, but increasing q leads to a Hopf bifurcation that produces an attracting limit cycle. When b = 1/3, b = 0.1, delta = 1, and q = 4, the ODE looks like: Using methods of Durrett (1993), it is not hard to show

Theorem. When the stirring rate is large there is coexistence.

A more interesting question, still under investigation, is to study the structure of the equilibrium state.

s3 Exercise. To see that there is something interesting to study run the model with its default parameter values.

Durrett, R. (1993) Predator-prey systems. Pages 37--58 in Asymptotic problems in probability theory: stochastic models and diffusions on fractals. Edited by K.D. Elworthy and N. Ikeda, Pitman Research Notes 83, Longman Scientific, Essex, England

Durrett, R., and Levin, S. (1997) Lessons from the planet WATOR. In preparation.