Analyzing multi-parameter bifurcation on a prey–predator model with the Allee effect and fear effect

In this research article, we present the dynamic analysis of the prey–predator model, adding the fear and the Allee effects. The main objective of this study is to analyze the periodicity and multi-parameter bifurcation of the model graphically. In particular, we studied the bifurcation between the two parameters. We critically analyzed the dynamics in the model with the help of detailed graphs of period ten and the Lyapunov exponent. Our study provides novel insights into the bifurcation behavior of the model, emphasizing the significance of Lyapunov exponents and the period-10 oscillation in understanding the dependencies among parameters. Through period ten oscillations, we confirm the complex dynamics, the coexistence of populations, and sensitivity to the initial conditions. Our analysis of the continuous-time model reveals that only Hopf bifurcation occurs at the positive fixed point, and we offer mathematical proof that no Hopf bifurcation occurs at other fixed points except the positive fixed point. From the numerical examples, we concluded that the crowding effect should be minimized for the stability of the model. Also, in the interior fixed point, when fear and Allee effects are taken as bifurcation parameters, backward bifurcations occur, which shows that in the presence of the crowding effect, the increase of the fear effect stabilizes the model.