Instability in a predator-prey system with delay and spatial averaging

Abstract

We consider a predator-prey reaction-diffusion system containing a term which, in its most general form, represents both spatial and temporal averaging. The eigenvalue equation arising from linearizing the system about a uniform coexistence steady state is studied using the argument principle to obtain a stability criterion, involving a delay measure that is not specific to a particular delay kernel but which works for a wide class of such kernels. In certain particular cases, stability criteria that cannot be improved are obtained using other techniques, and comparisons are made with the general bounds.