Marcus Garvie 
Florida State University

Numerical analysis of spatially-extended predator-prey systems
**************************************************************

We present the numerical analysis of two well-known reaction-diffusion
systems modeling nonlinear predator-prey interactions, where the local
growth of prey is logistic and the predator displays the Holling type II
functional response (the most frequently studied case). In a previous
study we proved the well-posedness of the classical solutions using
standard semi-group theory and a Lyapunov-type function. Numerical results
are presented for two fully-practical piecewise linear finite element
methods. We establish a priori estimates and error bounds for the
semi-discrete and fully-discrete finite element approximations. Numerical
results illustrating the theoretical results and spatiotemporal phenomena
(spiral waves and chaos) are presented in one and two space dimensions.
There are several implementational advantages of the finite element
methods, for example, they have equivalent finite difference
representations, and under mild restrictions of the time-step the
coefficient matrices of the resulting linear systems are strictly
diagonally dominant. The simplicity of the schemes means we need only 80
lines of Matlab code to solve a 2D problem with 2 million degrees of
freedom many thousands of times.