The recent work on this problem is explained and new results are offered. Weaker conditions are obtained to ensure global stability for dimension two, and a connection to the theory of neural networks is made. Using index theory and homotopy techniques, we study the existence of positive steady states of a class of reaction diffusion systems which general structure conditions. Reaction-diffusion equations describe interactions in a variety of systems in disciplines such as biology. An understanding of the behaviour of the systems is often drawn from qualitative analyses since analytical solutions are difficult to obtain.
Numerical approaches provide a key to discovering the behavior of the time dependent solutions.
They have been solved using an operator splitting method which can be readily extended for solving multi-species non linear reaction diffusion equations. The evolution of the biological systems and the steady state solutions in the form of travelling waves are accurately presented for each model. These solutions include phenomena such as waves of pursuit and evasion, competitive exclusion and cooperation for mutual advantange. The model for this study is a two-species discrete competitive system with two age classes, juveniles and adults. As the juveniles of one of the species mature, they disperse between two patches.