Study of Bifurcation and Delay-Driven Chaos in a Prey–Predator Model with Fear in Prey Reproduction and Two Forms of Harvesting

Abstract

In this paper, we delve into a predator–prey model incorporating a fear effect in prey reproduction, influenced by both delay and harvesting. The model accounts for delayed fear dynamics to capture more realistic dynamics. Initially, our focus lays on the nondelayed model, examining each biologically plausible equilibrium points and assessing their stability concerning the parameters of the model. Next, detailed mathematical results are provided, encompassing the asymptotic stability of all equilibria, Hopf bifurcation, and the direction and stability of bifurcated periodic solutions. Also, the stability analysis of the Hopf-bifurcating periodic solution is confirmed through the computation of first Lyapunov coefficient. Furthermore, we observed that the nondelayed system experiences Bogdanov–Takens bifurcation in a two-parameter space. Subsequently, we analyzed the corresponding delayed system, establishing the existence of a stable limit cycle through Hopf bifurcation concerning the delay parameter. Additionally, the inclusion of delay can prompt critical dynamics within the system, resulting in period-doubling routes toward chaotic oscillations. To validate our analytical findings, we conducted comprehensive and meticulous numerical simulations. The findings of the numerical simulations suggest that the impact of fear can be used as a measure of chaos control.