Numerical analysis of a first-order in time implicit-symplectic scheme for predator-prey systems

The numerical solution of reaction-diffusion systems modelling predator-prey dynamics using implicit-symplectic (IMSP) schemes is relatively new. When applied to problems with chaotic dynamics they perform well, both in terms of computational effort and accuracy. However, until the current paper, a rigorous numerical analysis was lacking. We analyse the semi-discrete in time approximations of a first-order IMSP scheme applied to spatially extended predator-prey systems. We rigorously establish semi-discrete a priori bounds that guarantee positive and stable solutions, and prove an optimal a priori error estimate. This analysis is an improvement on previous theoretical results using standard implicit-explicit (IMEX) schemes. The theoretical results are illustrated via numerical experiments in one and two space dimensions using fully-discrete finite element approximations.