Instability of Equilibria in Some Delay Reaction-Diffusion Systems

A new result is derived which extends a known instability result for a class of reaction-diffusion equations to a corresponding system incorporating time delay effects. For a significant class of nonlinear equations it is shown that an unstable equilibrium solution of the reaction-diffusion system cannot be stabilised by the introduction of delay. The result is applied to problems posed on convex domains with homogeneous Neumann boundary conditions. Finally, global methods in bifurcation theory are applied to a delay reaction-diffusion system of Lotka]Volterra type representing the interaction of two mobile species. The existence of a countable set of continua consisting of unstable equilibrium solutions for this system is proved.