Asian Research Journal of Mathematics

This study formulates and analyses a reaction-diffusion predator-prey model incorporating logistic prey growth, Holling type II predation, stage structure in the predator population, prey-taxis of adult predators, and a fixed maturation delay. The model is posed on a bounded spatial domain with homogeneous Neumann boundary conditions. Basic qualitative properties of the system are investigated, including non-negativity of solutions and an exponential bound for the total population under appropriate parameter restrictions. Three spatially homogeneous equilibria are identified: the extinction equilibrium, the prey-only equilibrium, and a coexistence equilibrium. A threshold quantity for predator invasion is derived using a next-generation matrix approach, and its dependence on key biological parameters is discussed through normalized sensitivity indices. Local stability is examined in both non-spatial and spatial settings. The analysis indicates that prey-taxis can influence the onset of spatial instability at the coexistence equilibrium, while the maturation delay may contribute to temporal oscillatory behaviour through a Hopf-type mechanism. Conditions for Turing instability and delay-induced oscillation are presented in terms of model parameters. A Lyapunov-Krasovskii functional approach is also used to discuss sufficient conditions for global stability of the equilibria. The results suggest that the combined effects of directed predator movement and predator maturation delay can produce qualitatively distinct dynamical regimes, including stable coexistence, stationary spatial patterns, temporal oscillations, and combined spatio-temporal dynamics. The findings provide a theoretical framework for understanding how stage structure, taxis-driven movement, and delay may interact in spatial predator-prey systems.