Durrett, R., and Levin, S.A. (2000) Lessons on
pattern formation from planet WATOR.
J. Theor. Biol. , to appear.
Postcript version
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 = 1/3, and q = 8, there is a limit cycle which is the smooth outer curve in the next figure:
Using methods of Durrett (1993), it is not hard to show
Theorem. When the stirring rate is large there is coexistence.
The next thing we want to understand the structure of the equilibrium state. The rough curve inside the ODE is a graph of the density of fish and sharks versus time in a 20 x 20 viewing window. To further investigate the process we will look at the behavior of densities in larger window sizes.
On to the NEXT PAGE or back to the HOME PAGE
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