ANALYSIS OF SPATIAL STRUCTURE IN A PREDATOR-PREY MODEL WITH DELAY .2. NONLINEAR-THEORY

A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for a predator-prey model including spatial diffusion and Volterra-type distributed delays in the interspecies interaction terms. For delays modeled by the ''weak'' generic kernel, the slow evolution of the amplitude of the spatially nonuniform states predicted by the linear analysis is shown to be governed by a complicated Ginzburg-Landau/Newell-Whitehead equation. Both the spatially-dependent and space-independent versions of this equation are analyzed to obtain the regimes of the physical parameter space where the linear nonuniform solutions either asymptote to a fixed amplitude wave pattern with an amplitude dependent frequency modulation, evolve to other permanent spatially-dependent wave solutions or patterns via nonlinear modulational instability, or decay to zero.