Optimal control of species augmentation in a competition model

Mathematical models of endangered competitive interactions incorporating the Allee effect with augmentation strategies have not been studied extensively. This area is however critical to ecologists since it relates to ways species can become endangered and possibly go extinct due to competition for limited resources. More importantly, the climatic change with its adverse effects has not only affected green forests but has also caused the extinction of some species. Thus, there is a need for critical augmentation strategies to safeguard such species. This paper, therefore, presents an optimal control strategy for a continuous time competition interaction model with strong Allee effects. We seek to maximize the target species at the end of each final time. We consider two objective functionals involving the populations and the cost of the controls. Using Pontryagin's Maximum Principle, we obtain the optimal control characterizations. We perform numerical simulations using the forward-backward sweep method and the approximate solutions are presented and discussed. Since there is a cost involved in the translocation of the reserve species, we adopt a minimization cost strategy. In addition, we compute the objective functional values for each simulation.