A lotka-volterra competition model with allee effect and stocking

This thesis consists of four main sections: an introduction and references, the main body, and the discussion and conclusion. In this study, a new model is developed based on the classical Lotka–Volterra competition model by incorporating the Allee effect for both species. In the proposed system, for the species sharing the same habitat and exhibiting a higher tendency toward extinction compared to the competing species, a stocking strategy proportional to the population size of the species introduced from outside the habitat is implemented. Furthermore, the nullclines of the derived model are determined, their intersection points are verified to obtain equilibrium points, and the qualitative analysis of these equilibria is performed. The results are then validated through numerical simulations based on this qualitative analysis. The fundamental corner equilibria of the model represent the absence of species or the persistence of a single species, and the stocking parameter S plays a critical role in the equilibrium structure, particularly through its effect on Species 1. The analysis demonstrates that stocking shifts the nullcline of Species 1 upward, thereby altering the locations of the interior equilibria. Regarding the stability properties of the equilibrium points, it is shown that the stability of the corner equilibria associated with Species 2, which is not subjected to stocking, may differ from those in the classical Lotka–Volterra competition model with Allee effects for specific choices of the stocking parameter. Although the stability of the corner equilibria corresponding to Species 1 remains unchanged, the stability of the trivial equilibrium (0,0) is affected by the relative magnitude of the stocking parameter with respect to A_1 r_1(The product of the Allee threshold value and the growth rate of the stocked species). The stability of the interior equilibrium points remains unchanged, consistent with the Lotka–Volterra competition model incorporating Allee effects. Phase portraits and heat maps obtained from numerical simulations confirm the analytical results and visualize species extinction, persistence below the Allee threshold, or stabilization near the carrying capacity depending on the value of the stocking parameter. While stocking facilitates Species 1 in overcoming the Allee threshold and induces a temporary increase in population size in the short term, environmental limitations ensure that the carrying capacity remains unchanged in the long term. In conclusion, the Lotka–Volterra competition model incorporating both stocking and Allee effects is investigated using analytical and numerical approaches. Critical stocking thresholds are identified, and their decisive roles in determining coexistence or extinction scenarios are clearly demonstrated. This study shows that stocking can be a powerful tool for species conservation; however, if the presence of Allee effects in both species is neglected, stocking programs may disrupt ecosystem balance and potentially lead to the extinction of certain species. Therefore, stocking strategies should be designed by considering not only the target species but also the Allee thresholds and carrying capacities of competing species. In this way, the present study highlights, within a mathematical framework, the importance of stocking practices in maintaining ecological balance and ensuring species sustainability.