Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator–prey model with additional food for predators

Abstract

In biological control programs, the provision of additional food is regarded as one of the most useful tools for species conservation and pest management. In a system, providing additional food for predators can distract the predators from the overconsumption of prey (short term) or enhance the predation rate (long term). In the present article, we consider a predator–prey model with Holling type-II functional response by incorporating additional food for the predator species in a discrete-time setup. Here, we intend to explore the impact of additional food on the growth of prey as well as the overall dynamics of the underlying system. We analyze the system dynamics by varying two control parameters, viz, the biomass of additional food and the intrinsic growth rate of prey species simultaneously with the help of Lyapunov exponent and isoperiodic diagrams. Through the simultaneous variation of two parameters, very rich and complex dynamical behaviors of a simple deterministic system can come into view with the presence of structurally stable periodic patterns in the transitional and chaotic zones, which is not looked on only by the variability of one parameter. In this study, we notice the presence of infinite families of different organized periodic structures, like Arnold tongues, shrimp-shaped domains, a new kind of ‘double fishhook’ structures, etc. in the quasiperiodic and chaotic zones of the parameter planes with various kinds of period-adding phenomena. The study also discloses the transition to chaos via shrimp-induced period-bubbling process. We observe the coexistence of double and triple attractors in several different sets of attractors. The most fascinating finding in this study is the coexistence of quadruple attractors, one of the rarest phenomena in ecological systems. The basin boundaries of all these coexisting double, triple, and quadruple attractors are of a very complex nature, they are either fractal basin boundaries or Wada basin boundaries.