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 E₀, predator-free coexistence E₁, and interior coexistence E₂. The existence of E₁ follows from an application of the Intermediate Value Theorem to a cubic polynomial, while the coexistence equilibrium E₂ is expressed in explicit closed form for each component. The basic reproduction number R₀ 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 d₂ each carry a sensitivity index of magnitude one, making them the dominant controls on R₀, whereas the delay-mortality product d₁ τ ranks as the next most influential quantity. Local asymptotic stability of E₁ and E₂ is determined through explicit Routh-Hurwitz conditions on the characteristic quasi-polynomial. Global asymptotic stability of E₁ when R₀1 is established using Lyapunov-Krasovskii functionals. The model undergoes a transcritical bifurcation at R₀=1, and a Hopf bifurcation of E₂ arises when τ surpasses the critical threshold τ₀^*, whose determination reduces to locating positive roots of a scalar cubic.
Wafula et al. (Wed,) studied this question.
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