Local stability and Hopf Bifurcation analysis of a delayed Three-Prey Two-Predator nondiffusive system

Abstract

This study explored how time delays affect the stability and behavior of a predator–prey ecosystem involving three prey and two predator species. Building on earlier models proposed by a delay term is introduced to represent the predators’ gestation or handling time, using a generalized Lotka–Volterra framework. Each prey population grows logistically, while both predators feed on all three prey species through a linear functional response. The resulting five-dimensional delay differential model is analyzed through linearization and the Routh–Hurwitz stability criterion for the case without delay. From the Jacobian matrix evaluated at the interior equilibrium, we derive the characteristic equation and examine the sys tem’s local stability. The analysis showed that the equilibrium point remains locally asymptotically stable as long as all eigenvalues of the Jacobian have negative real parts. Numerical simulations supported these theoretical findings, demon strating that the introduction of delay can destabilize an otherwise stable equilibrium, resulting in periodic population cy cles similar to those observed in natural ecosystems. When the delay exceeded a critical threshold (τₐ ≈ 1.7), a pair of com plex conjugate eigenvalues crosses the imaginary axis, triggering a Hopf bifurcation and giving rise to sustained oscilla tions. The study underscores the key role of time delays in shaping predator–prey dynamics and offers new insights into how delayed feedback influences ecological stability, persistence, and coexistence.