Approximated Spiral and Target Patterns in Bazykin’s Prey–Predator Model: Multiscale Perturbation Analysis

Abstract

Spatio-temporal pattern formation has opened up a wide area of research to understand the dynamics between various interacting populations such as a prey and a predator, competing species etc. The governing equations are typically modeled by a reaction–diffusion system. The commonly known patterns, namely, traveling wave, periodic traveling wave, spot, labyrinthine, mixture of spot and stripe, spatio-temporal chaos and interacting spiral chaos can be observed in the spatio-temporal extension of various interacting population models. Apart from these, the other two types of patterns, namely, spiral and target patterns also evolve under suitable parametric conditions near the Turing–Hopf threshold though there is no systematic approach to determine the exact formalism of their emergence. In this paper, we have used a multiscale perturbation analysis to determine these patterns in the spatio-temporal extension of Bazykin’s prey–predator model. An important finding of this work is the use of approximated analytical solution as the initial condition to obtain spiral and target patterns with the help of numerical simulations. Analytical results are general in nature and hence can be used for any spatio-temporal model of interacting population as well as other pattern forming systems which are capable of producing spiral and target patterns.