Complex dynamics in a slow–fast modified Leslie–Gower model: the effects of predator migration under central place foraging framework

Abstract

This article investigates the dynamics of a predator–prey system incorporating central place foraging (CPF) framework. This framework is structured into two distinct spatial zones: a foraging ground (zone 1), inhabited by both prey and a fraction of the predator population, and a breeding colony (zone 2) exclusively for the remaining predators. The model incorporates a prey-density-dependent migration pattern of predators. The system utilizes a Beddington–DeAngelis functional response for predator–prey interactions and follows a modified Leslie–Gower scheme to capture net demographic recruitment. The model formulation is governed by a three-dimensional system of ordinary differential equations featuring slow–fast dynamics, reflecting that the predator’s spatial commuting occurs on a much faster time scale compared to demographic growth and interaction. To analyze the complete three-dimensional slow–fast system, the method of aggregation is used, reducing it to a two-dimensional, single-time-scale system. The fundamental properties of this aggregated system are established, including positivity and boundedness. Additionally, conditions for the existence of equilibrium points are derived, and their local stability is investigated through linear stability analysis of the Jacobian matrix. An in-depth bifurcation analysis is performed to understand qualitative behavioral regime shifts. Several codimension-one bifurcations are observed, including a transcritical bifurcation near the prey-free equilibrium, as well as saddle-node and Hopf bifurcations at the coexistence equilibrium. Additionally, the system experiences stability switching near the coexistence equilibrium through successive Hopf bifurcations. As a result, critical thresholds are identified for both population collapse and sustained periodic solutions. Furthermore, a two-parameter continuation reveals rich and complex topological structures, including Bogdanov–Takens, generalized Hopf, and cusp bifurcations. All analytical findings are validated through numerical simulations using MATLAB and MatCont, demonstrating that the interaction of spatial commuting constraints and population dynamics generates highly complex, non-linear ecological behavior.