DYNAMICS OF A PREY–PREDATOR MODEL: UNTANGLING THE ROLE OF FEAR-INDUCED ALLEE EFFECT, PREY REFUGE, AND GROUP DEFENSE

Ecological systems often exhibit complex dynamics due to interactions between predator and prey species, which can be influenced by both direct predation and indirect effects like fear and population thresholds. We introduce a predator–prey model that incorporates non-consumptive effects such as the Allee effect, group defense, and prey refuge. These non-consumptive effects are triggered by predator fear, along with a predator-taxis sensitivity parameter. We explore interactions between fear and defense mechanisms. Our ODE model’s comprehensive mathematical analysis addresses the positivity and boundedness of solutions, species extinction, and the existence and stability of equilibria, including various local bifurcations. We present a one-parameter bifurcation analysis focusing on the Allee parameter and predator-taxis sensitivity. This analysis reveals multiple interior equilibrium points with varying stability, occurrences of saddle-node, transcritical, and Hopf bifurcations, as well as instances of bistability and tristability. We find that a strong Allee effect initially supports species coexistence through tristability. However, beyond a certain threshold, the Allee parameter leads to unsustainable coexistence. Similarly, initial predator sensitivity may encourage coexistence through multiple stable states, but excessive sensitivity makes coexistence infeasible. The outcomes — ranging from complete extinction to predator extinction — depend critically on initial population densities. Further, we delve into the spatial dynamics of the associated reaction–diffusion prey–predator interaction model that incorporates cross-diffusion effects. We establish conditions for stability and Turing instability, validate through numerical simulations. These simulations reveal diverse spatial patterns such as irregular mixtures of spots, and cluster of spots like patterns that underscore the influence of time iteration, cross-diffusion coefficients and the Allee threshold on spatial density distribution of species.