Spatial structures in predator-prey communities with hereditary effects and diffusion

Abstract

We study a continuous time delay interaction model for predator-prey communities in which the species can also diffuse. Oscillatory traveling wave solutions are obtained. The amplitudes (and also the wavenumbers and frequencies) are seen to evolve slowly in time and to vary significantly over large regions of space due to the effects of small nonlinear terms in the model. These modulations (amplitude and dispersion relation) are found in terms of the parameters of the problem. Other authors have conjectured that these waves (which are nonlinear perturbations of linear waves) are stable, based on various restricted stability analyses. Based on a full stability analysis, we show that all such waves (both steadily progressing and steady solutions) are unstable. We conclude that periodic bifurcations from an equilibrium state are unstable, but periodic structure bifurcating from a finite amplitude spatially homogenous limit cycle can be stable.

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