Abstract
Predator-prey dynamics are strongly influenced by environmental factors, such as the availability of resources, habitat quality and climatic conditions, which determine species survival, species reproduction and aggregate defensive behaviour. Using these ecological processes, a three-species predator-prey model is formulated and analysed, with prey group defence dependent on an environmental logistic function. Predation intensity, influenced by environmental conditions, is examined, and the long-term dynamics of interacting populations under these conditions are explored. Biologically plausible equilibrium points are verified, and their local and global stability is thoroughly investigated by the Routh-Hurwitz criterion and Lyapunov function methods, respectively. The governing equations are modified to include diffusion terms to describe the dispersal of species in space, leading to a reaction-diffusion system that describes the spread of species in heterogeneous habitats. A three-level finite difference scheme is developed to approximate the diffusive model, and its numerical stability is examined through Von Neumann analysis. Furthermore, the stability of the reaction-diffusion system is investigated using a Fourier series expansion and the Routh-Hurwitz criterion. Increased effective predation, larger population oscillations, and less stable coexistence between species are shown to result from a reduction in the strength of environmentally regulated group defence. In contrast, diffusion can help smooth spatial population gradients, making the ecosystem more resilient to effective predation and promoting species redistribution. The mathematical model proposed here can contribute to a better understanding of the complex interplay between environmental drivers, collective defense, and spatial dispersion in predator-prey dynamics and is potentially applicable to ecological modelling, biodiversity conservation, habitat management, and the analysis of spatially structured ecosystems.