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Foraging movements of predator play an important role in population dynamics of prey-predator systems, which have been considered as mechanisms that contribute to spatial self-organization of prey and predator. In nature, there are many examples of prey-predator interactions where prey is immobile while predator disperses between patches non-randomly through different factors such as stimuli following the encounter of a prey. In this work, we formulate a Rosenzweig-MacArthur prey-predator two patch model with mobility only in predator and the assumption that predators move towards patches with more concentrated prey-predator interactions. We provide completed local and global analysis of our model. Our analytical results combined with bifurcation diagrams suggest that: (1) dispersal may stabilize or destabilize the coupled system; (2) dispersal may generate multiple interior equilibria that lead to rich bistable dynamics or may destroy interior equilibria that lead to the extinction of predator in one patch or both patches; (3) Under certain conditions, the large dispersal can promote the permanence of the system. In addition, we compare the dynamics of our model to the classic two patch model to obtain a better understanding how different dispersal strategies may have different impacts on the dynamics and spatial patterns.

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We consider a system of delay differential equations to represent predator-prey eco-epidemic dynamics with weak Allee effect in the growth of predator population. The basic aim of the paper is to observe the dynamics of such system under the influence of gestation delay of predator and Allee parameter. We analyze essential mathematical features of the proposed model such as uniform persistence, stability and Hopf-bifurcation at the interior equilibrium point of the system. Global asymptotic stability analysis of the positive equilibrium points by constructing a suitable Lyapunov function for the delayed model is carried out separately. We perform several numerical simulations to illustrate the applicability of the proposed mathematical model and our analytical findings. We observe that the system exhibits chaotic oscillation due to increase of the delay parameter τ. We also observe that there is a threshold of Allee parameter above which the predator population will be washed away from the system.