Abstract
We consider a mathematical model to describe the interaction between predator and prey that includes predator cannibalism and refuge. We aim to study the dynamics and its long-term behavior of the proposed model, as well as to discuss the effects of crucial parameters associated with the model. We first show the boundedness and positivity of the solution of the model. Then, we study the existence and stability of all possible equilibrium points. The local stability of the model around each equilibrium point is studied via the linearized system, while the global stability is performed by defining a Lyapunov function. The model has four equilibrium points. It is found that the equilibrium point representing the extinction of both prey and predator populations is always unstable, while the other equilibrium points are conditionally stable. In addition, there is forward bifurcation phenomena that occur under certain condition. To support our analytical findings, we perform some numerical simulations.
Highlights
Predator–prey interaction is one of the most important issues in ecology, as it is the basis of the food chain
Many mathematical models have been proposed in the literature to understand the dynamics of predator–prey interaction
The mathematical model of cannibalism has been studied by some researchers [4–6]
Summary
Predator–prey interaction is one of the most important issues in ecology, as it is the basis of the food chain. Zhang et al [5] analyzed predator–prey models with cannibalism and stage structure in predators so that the model studied was a three-dimensional dynamical model. Predation of prey and juvenile predators by adult predators follows the type I Holling functional response. Deng et al [6] studied a two-dimensional predator–prey model with predator cannibalism: N dN. Many prey species adopt the technique of refuge to avoid the predation. The mathematical model of predator–prey with prey refuge has been widely studied [16–18]. We propose a model describing the predator–prey interaction incorporating predator cannibalism and refuge and perform a dynamical analysis for the proposed model. The proposed model is a development of Deng’s model [6], namely by implementing the Holling type II functional response instead of the Holling type I and assuming that there is predator refuge from cannibalization.
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