Resonant bifurcations and complex dynamics in a coupled logistic prey–predator map with proportional immigration

Abstract

We investigate a discrete-time predator-prey model that combines logistic prey growth, bilinear predation, and density-dependent proportional immigration into the prey population. We first identify the invariant coordinate axes and, under suitable parameter conditions, construct a positively invariant trapping region for the dynamics. All fixed points are then determined, and explicit local stability conditions are derived for the boundary and coexistence equilibria. At the coexistence equilibrium, we obtain analytical criteria for transcritical, period-doubling, and Neimark-Sacker bifurcations. The two-parameter bifurcation structure is organized by several resonant configurations. We identify a 1:2 codimension-two point at the intersection of the period-doubling and Neimark-Sacker curves, and we analyze additional resonance points along the Neimark-Sacker set, including the strong resonances 1:3 and 1:4, together with the weak 1:5 resonance. These resonances generate a rich local organization of the parameter plane, involving invariant closed curves, phase-locking regions, periodic windows, and transitions to chaotic dynamics. Basin computations further reveal multistability near the 1:3 resonance and a detailed phase-space organization near the 1:4 resonance, including attracting and saddle period-four structures under the fourth iterate. The results show that proportional prey immigration substantially reshapes the bifurcation geometry of the coupled logistic map and promotes a broad spectrum of asymptotic behaviors, from stable coexistence to quasiperiodicity, resonant periodicity, and chaos.