## 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[1] 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[2].
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] = 1/3, b[2] = 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.*

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