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Stability and Bifurcation Patterns in Multi-Species Predator–Prey Systems

Kamaljeet Kaur, Assistant Professor, Guru Nanak College, Sri Muktsar Sahib

Abstract

Mathematical models of three-species predator-prey systems are essential for understanding complex ecological dynamics, as these 3D continuous models represent the minimum dimension capable of exhibiting deterministic chaos. This paper reviews the analytical methodologies and dynamical implications associated with stability and bifurcation in tri-trophic food chains and intra-guild predation (IGP) systems. Local asymptotic stability of coexisting equilibrium points (E*) is rigorously assessed using the Routh–Hurwitz criterion on the characteristic polynomial of the Jacobian matrix, an algebraic test critical for analyzing highly nonlinear systems without explicit root calculation. Bifurcation analysis reveals how parameter variations drive qualitative shifts in dynamics: Transcritical bifurcations govern species invasion/extinction, while Hopf bifurcations generate self-sustained population oscillations (limit cycles). In discrete systems, the analogous Neimark–Sacker bifurcation leads to quasi-periodic behaviour. A key challenge involves the destabilizing influence of ecological modifications, such as gestation time delays, which induce stability switches by repeatedly moving the characteristic equation roots across the imaginary axis. Conversely, realistic non-consumptive effects like prey refuge and adaptive foraging behaviour are shown to be powerful stabilizing forces, capable of collapsing chaotic dynamics to stable states, provided critical trophic efficiency conditions are met. Advanced geometrical methods, such as singular perturbation analysis, demonstrate that deterministic Shilnikov chaos is structurally guaranteed under explicit conditions of extreme time-scale separation (ζ≪1) and high trophic efficiency (ϵ>ϵ0​ of the top predator). This synthesis underscores the delicate balance between structural realism and analytical tractability in mapping complex ecological interactions to stable or unstable outcomes.

Keywords: Predator-Prey, Bifurcation, Routh-Hurwitz, Tri-Trophic, Chaos, Stability.