We develop and analyze a reaction–diffusion prey–predator model that integrates predator-induced fear, constant prey refuge, and cooperative interactions among specialist predators. To capture realistic density regulation, intraspecific competition within the prey population is modelled using Richards-type growth, while predator harvesting is incorporated as an external ecological control mechanism.
The dynamical behaviour of the associated non-spatial system is first examined, revealing the presence of multiple equilibria, namely the trivial equilibrium, the predator-free (axial) equilibrium, and a biologically feasible coexistence equilibrium. Local stability analysis demonstrates that the persistence of the coexistence state is strongly governed by critical ecological parameters, including handling time, fear intensity, and harvesting rate. Variations in these parameters induce qualitative shifts in system dynamics, and the coexistence equilibrium is shown to undergo Hopf bifurcation, resulting in the emergence of stable and oscillatory solutions. The corresponding threshold values are determined through numerical continuation techniques implemented in MATLAB and MATCONT.
The model is subsequently extended to a spatial framework to explore diffusion-driven instability. Through combined analytical investigation and numerical simulations, we establish conditions for the onset of Turing instability and the formation of self-organized spatial patterns. The results underscore the significant influence of diffusion rates and biological interactions on species dispersal and spatial organization. In particular, the dispersion analysis indicates that Turing patterns arise when the prey diffusion rate exceeds that of the predator, highlighting the role of differential mobility in pattern formation.
Overall, this work presents a biologically consistent modelling framework that elucidates the interplay between ecological interactions and spatial processes, offering new insights into the mechanisms underlying complex spatiotemporal dynamics in prey–predator systems.
