Dynamical Analysis of a Predator-Prey Model Incorporating Predator Cannibalism and Refuge

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.