Delay-induced Hopf bifurcation and stability transitions in a nonlinear spatially coupled two-patch system

This study investigates delay-induced stability transitions in a nonlinear two-patch population system with density-dependent passive dispersal. The model is formulated as a system of retarded functional differential equations in which current emigration is modulated by past source-patch density, while immigration depends on delayed movement from the opposite patch. Local existence, uniqueness, and positivity of solutions are established, and the coexistence equilibrium conditions are derived. For a representative parameter set, a positive coexistence equilibrium is computed and shown to be locally asymptotically stable in the absence of delay. Trace–determinant conditions characterize the nondelayed stability, while an explicit transcendental characteristic equation is derived for the delayed system. A simple pair of characteristic roots crosses the imaginary axis at a critical delay of 3.7367, with an associated Hopf frequency of 0.7555. The positive transversality coefficient confirms that the equilibrium loses stability as the delay increases through the critical value. A model-specific center-manifold and normal-form calculation gives a first Lyapunov coefficient of − 0.05334, confirming a supercritical Hopf bifurcation and orbitally asymptotically stable periodic solutions for delays slightly above the threshold. Long-time simulations and an adaptive amplitude-versus-delay analysis provide computational support for the analytical classification by showing damped convergence below the threshold and stable periodic coexistence above it. A one-at-a-time sensitivity analysis using ±10% perturbations of all seven model parameters shows that the Hopf transition persists throughout the investigated parameter neighborhood, with positive transversality retained in all perturbed cases. The critical delay is particularly sensitive to the strength of density-dependent dispersal feedback. Additional nondelayed simulations show that unequal carrying capacities redistribute equilibrium abundance between the two patches without reproducing the delay-induced oscillatory transition. These results demonstrate the local robustness of the delay-induced Hopf mechanism and distinguish the equilibrium effects of habitat heterogeneity from the stability effects of delayed dispersal feedback.