Mathematical Analysis of a Stage-Structured Two-Patch Predator-Prey Model with Density-Dependent Prey Migration and Predator Maturation Delay

We propose and analyse a stage-structured predator-prey model in which prey are distributed across a refuge patch and a predation patch, with density-dependent fear-driven movement between them, while the predator population develops through two distinct life stages separated by a fixed maturation delay τ. Predation acts exclusively on exposed prey according to a Holling type II functional response. Non-negativity of solutions is established through quasi-positivity of the vector field, and exponential boundedness is demonstrated via Gronwall’s inequality. The model admits three biologically meaningful equilibria: total extinction E_0, predator-free coexistence E_1, and interior coexistence E_2. The existence of E_1 follows from an application of the Intermediate Value Theorem to a cubic polynomial, while the coexistence equilibrium E_2 is expressed in explicit closed form for each component. The basic reproduction number R_0 is obtained by applying the next-generation matrix method with a delay-adjusted survival factor. Normalised sensitivity indices are derived analytically for every parameter; the conversion efficiency β and adult predator mortality d_2 each carry a sensitivity index of magnitude one, making them the dominant controls on R_0, whereas the delay-mortality product d_1 τ ranks as the next most influential quantity. Local asymptotic stability of E_1 and E_2 is determined through explicit Routh-Hurwitz conditions on the characteristic quasi-polynomial. Global asymptotic stability of E_1 when R_0<1 and of E_2 when R_0>1 is established using Lyapunov-Krasovskii functionals. The model undergoes a transcritical bifurcation at R_0=1, and a Hopf bifurcation of E_2 arises when τ surpasses the critical threshold τ_0^*, whose determination reduces to locating positive roots of a scalar cubic.