TURING INSTABILITY IN COMPETITION MODELS WITH DELAY .1. LINEAR-THEORY

Turing instability in two-component predator-prey and reaction-diffusi on models including diffusion and Volterra-type distributed delays in the interspecies interaction terms is considered. For general function al forms of the prey birthrate-predator deathrate/reaction terms and d elays modeled by the ''weak'' generic kernel a exp(-aU) or the ''stron g'' generic kernel aU exp(-aU), the structure of the diffusively-unsta ble space is shown to be completely altered by the inclusion of delays . The necessary and sufficient conditions for Turing instability are d erived using the ''weak'' generic kernel and are found to be significa ntly different from the classical conditions with no delay. The struct ure of the Turing space, where steady states may be diffusionally driv en unstable initiating spatial pattern, is delineated for several spec ific models, and compared to the corresponding regimes in the absence of delay. An alternative bifurcation-theoretic derivation of the bound ary of the Turing-unstable domain is also presented. Finally, the inst ability with delays is briefly considered for two spatial dimensions a nd a finite domain size.