The persistence and bifurcation analysis of an ecological model with fear effect involving prey refuge and harvesting

In this work, the sufficient conditions for uniformly persistence of a food chain prey-predator model with prey refuge and harvesting have been found. By creating suitable Lyapunov functions, a set of sufficient conditions that can be checked easily are gotten for the global asymptotic stability of the model. The conditions of the local bifurcation are determined, it’s noticed that near axial and free-second predator equilibrium points there is a transcritical bifurcation, otherwise near the free-second predator equilibrium point there is a pitchfork bifurcation, while near the coexistence equilibrium point there is a saddle-node bifurcation . Additional realizations for the Hopf bifurcation close to coexistence equilibrium point have been done. The validity of our main results was demonstrated by numerical analysis.