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In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments. [ABSTRACT FROM AUTHOR]

To reduce the chance of predation, many prey species adopt group defense mechanisms. While it is commonly believed that such defense mechanisms lead to positive feedback on prey density, a closer observation reveals that it may impact the growth rate of species. This is because individuals invest more time and effort in defense rather than reproductive activities. In this study, we delve into a predator–prey system where predator-induced fear influences the birth rate of prey, and the prey species exhibit group defense mechanism. We adopt a nonmonotonic functional response to govern the predator–prey interaction, which effectively captures the group defense mechanism. We present a detailed mathematical analysis, encompassing the determination of feasible equilibria and their stability conditions. Through the analytical approach, we demonstrate the occurrence of Hopf and Bogdanov–Takens (BT) bifurcations. We observe two distinct types of bistabilities in the system: one between interior and predator-free equilibria, and another between limit cycle and predator-free equilibrium. Our findings reveal that the parameters associated with group defense and predator-induced fear play significant roles in the survival and extinction of populations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear ratio-dependent prey–predator model with constant prey refuge in the prey population. Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population. The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail. Hopf bifurcation including its direction and stability for the model is also studied. We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system. It is also observed that for a fixed strength of fear, an increase in the Allee parameter makes the system unstable, whereas an increase in prey refuge drives the system toward stability. However, higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction. Further, we explore the variation of densities of the populations in different bi-parameter spaces, where the coexistence equilibrium point remains stable. Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software. [ABSTRACT FROM AUTHOR]