BIFURCATION ANALYSIS OF A DISCRETE PREDATOR–PREY MODEL WITH FEAR EFFECTS, PREY REFUGE, HARVESTING AND CROWLEY–MARTIN FUNCTIONS

In this paper, we consider a discrete predator–prey model with fear effects, prey refuge, harvesting and Crowley–Martin functional responses. First, we analyze the existence of the model’s fixed points. Subsequently, we apply bifurcation theory to establish the conditions for the occurrence of codimension-two bifurcations, including fold-flip bifurcation, 1:1 strong resonance and 1:2 strong resonance. Finally, we verify our theoretical analysis through numerical simulations, presenting bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and the two-parameter spaces under the specified initial conditions.