Mappings of the plane that simulate prey-predator systems
A prey-predator system that is spatially dependent is built around the familiar logistic equation. The result is a set of coupled integro-differential equations. If at least one of the populations reproduces periodically in a time that is short to other characteristic times, these equations can be transformed to a set of coupled maps, each map representing a spatial point. Various models are developed that include predator ranging as well predator migration via diffusion. In two dimensions the coupling among the spatial points will have the symmetry of the regular polyhedra if the infinite plane is viewed in polar coordinates, or toroidal symmetry if periodic boundary conditions are imposed. Numerical examples are given. Depending on the value of the parameters and on the initial conditions the system can oscillate in a variety of different spatial modes, giving rise to unusual bifurcation portraits, or it can behave chaotically. The solutions show that a patchy distribution of population is more stable against the influences of environmental noise than is a smoothly distributed population. The numerical results in two dimensions are in constrast with the results of a previous study in one dimension.