Structure of parameter space of a three-species food chain model with immigration and emigration

Migration is a natural behavior and an integral part of many species' life cycles. Although most commonly found in many species of mammals and birds, it also occurs in several other species such as fish, insects, etc. Animals migrate in response to the spatial and temporal variability of environmental factors, such as food availability, habitat safety, climate, and mating opportunities. The present study investigates the role of middle predator's migration (immigration and emigration) in the dynamics of a well-known tri-trophic food chain model. We perform extensive numerical simulations of this model system with simultaneous variation of migration and another system parameter related to the half-saturation constant of the middle predator, and present a collection of high-resolution isospike and Lyapunov exponent diagrams drawn in the biparametric space illustrating the intricate nature of the system dynamics. We mainly find that a moderate amount of migration (both immigration and emigration) promotes regularity in the dynamics of the system. High migration rates, however, lead to the extinction of one or more species from the system. The isospike diagrams uncover several periodic windows of different periodicity inside the chaotic region, some of them crossing one another. We demonstrate with the aid of phase portraits and basins of attraction that these overlappings induce bistability between coexisting attractors. We notice that these basins have a self-similar nature. Additionally, the system exhibits shrimp-shaped periodic structures, period-bubbling route to chaos, and multiple-times stability switching. We also include several animations related to stability switching and the basin of attraction for better visualization of the dynamics of the system.