Unveiling complex dynamics and control: Codimension-2 bifurcations and time-delay feedback control in a discrete predator–prey model

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To understand and predict complex dynamical behavior especially when codimension-1 bifurcations are insufficient to capture a full range of phenomena for a system, unfolding codimension-2 bifurcations help predict secondary bifurcations, multistability, and transitions to chaos. This article examines the dynamics of a modified discrete predator–prey model incorporating mixed functional responses. Stability analysis via linearization elucidates the long-term behavior of this ecological system, predicting population persistence, extinction, or fluctuations to inform effective ecosystem management. The principal aims are to investigate codimension-2 bifurcations related to strong resonances (1:2, 1:3, and 1:4), secondly to derive parametric conditions for these resonances at interior (positive) fixed point, and to develop a chaos control strategy for ecological stabilization. Rigorous bifurcation analysis establishes resonance conditions, validated through numerical simulations illustrating qualitative transitions near resonance points. Furthermore, we implement a time-delay feedback control (TDFC) method that successfully suppresses chaotic dynamics by incorporating biologically feasible interventions. Numerical results demonstrate effective stabilization to sustainable population levels using minimal control effort, providing a mathematical framework for ecosystem management.

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