Pattern Formation in Bazykina’s Predator-Prey Model with Holling Type-III Response: A Multiscale Perturbation Analysis

Abstract

The dynamics of interacting populations—such as rival species or predator-prey systems—can be more thoroughly understood through the study of spatiotemporal pattern formation. Reaction-diffusion systems are commonly used to model such interactions. These models exhibit a wide range of well-known patterns, including travelling waves, periodic travelling waves, spots, labyrinthine structures, mixed spot-stripe patterns, spatiotemporal chaos, and interacting spiral chaos. Under appropriate parametric conditions, both target and spiral wave patterns can emerge. Notably, spiral patterns have been observed to evolve at the Turing-Hopf threshold; however, the precise formalism responsible for their emergence has remained unclear. This study employs a multiscale perturbation analysis to elucidate the mechanisms behind spiral pattern formation under suitable parametric regimes. A key contribution of this work is the discovery that spiral patterns can be generated by initiating numerical simulations from approximate analytical solutions. The behavioral insights derived from this study are broadly applicable to a wide class of pattern-forming systems that exhibit spiral structures, as well as to general spatiotemporal models of interacting populations.