This study improves our comprehension of the prey-predator system while potentially having implications for other complex systems in various fields, including population dynamics, epidemiology, and economics. To prevent bifurcation, we employed a simple control method. While the decrease of these two effects causes an increase in growth rate, which causes bifurcation in the system due to overcrowding, the addition of the Allee effect and the fear effect should be, to a certain extent, so that the excess on both impacts controls the crowding effect. Similarly, the more significant Allee effect stabilizes 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. From the numerical examples, we concluded that the crowding effect should be minimized for the stability of the model. 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. Through period ten oscillations, we confirm the complex dynamics, the coexistence of populations, and sensitivity to the initial conditions. 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. We critically analyzed the dynamics in the model with the help of detailed graphs of period ten and the Lyapunov exponent. In particular, we studied the bifurcation between the two parameters. The main objective of this study is to analyze the periodicity and multi-parameter bifurcation of the model graphically. In this research article, we present the dynamic analysis of the prey-predator model, adding the fear and the Allee effects.
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