Allee effects in continuous-time and discrete-time predator-prey system

In this thesis, the dynamical behaviour of both discrete-time and continuous-time Leslie type predator-prey system with the Allee effect is investigated. These models has two populations which are prey and predator living in the same environment and interacting with each other. In these systems the change of populations, modeled by approximation of nonlinear dynamical systems, with respect to time is governed by difference equations in discrete system and by differential equations in continuous system. First, the discrete-time prey-predator system is considered and the existence and uniqueness of the positive equilibrium point of this system is shown. The required conditions are then found for the stability of this positive equilibrium point, as well as for the observation of Flip bifurcation and Neimark-Sacker bifurcation at this equilibrium point. More specifically, these bifurcations are driven by using the center manifold theorem and the normal form theory by choosing the integral step size as a bifurcation parameter. In the following chapter, some numerical simulations are presented to support and extend the theoretical results. The other half of the thesis is devoted to the continuous-time predator-prey system, where the discrete system is obtained using the Euler method. The continuous system follows the same sequence of basic analyses as the discrete system. Initially, the local stability conditions of the positive equilibrium point of this system are determined. Next, the conditions of existence for Hopf bifurcation arising from this positive equilibrium point are investigated. This bifurcation is demonstrated through Hopf bifurcation theory and normal form theory by using the Allee constant as a bifurcation parameter. According to mathematical analysis, the Allee effect reduces prey population density since prey have difficulty finding a mate to reproduce. In addition to this, since prey populations are the primary nutritional source for predator populations, the Allee effect diminishes predator population density. On the other hand, at the lower level of population density, we have observed that a gradually increasing Allee effect causes a supercritical Hopf bifurcation, while a strong Allee effect leads to a subcritical Hopf bifurcation. To put it another way, there is a stability switch from stable to unstable and from unstable to stable as the Allee effect increases. Therefore, the system exhibits multiple Hopf bifurcations since the Allee effect can bring out both stable and unstable limit cycles. Subsequently, we focus on a sensitivity analysis to ascertain the robustness of the model to the parameter values that are correlated with the critical bifurcation parameters directly related to the Allee constant. Finally, we discuss the impact of the Allee effect on the dynamics of both prey and predator populations via numerical simulations. Moreover, the FAST approach was used to examine the sensitivity of the predator-prey system to all parameter values, and the most influential factors among input parameters on the output variable have been determined in addition to the impact of a single parameter value change on the dynamic of the system. The findings highlight the importance of the Allee effect in maintaining the intricate balance between prey and predator populations and the importance of considering complex ecological interactions in order to accurately model and understand these systems.