Turing Instability in Competition Models with Delay I: Linear Theory

Abstract

Turing instability in two-component predator-prey and reaction-diffusion models including diffusion and Volterra-type distributed delays in the interspecies interaction terms is considered. For general functional forms of the prey birthrate-predator deathrate/reaction terms and delays modeled by the “weak” generic kernel $a\exp ( - aU )$ or the “strong” generic kernel $aU\exp ( - aU )$, the structure of the diffusively-unstable space is shown to be completely altered by the inclusion of delays. The necessary and sufficient conditions for Turing instability are derived using the “weak” generic kernel and are found to be significantly different from the classical conditions with no delay. The structure of the Turing space, where steady states may be diffusionally driven unstable initiating spatial pattern, is delineated for several specific models, and compared to the corresponding regimes in the absence of delay. An alternative bifurcation-theoretic derivation of the boundary of the Turing-unstable domain is also presented. Finally, the instability with delays is briefly considered for two spatial dimensions and a finite domain size.