Numerical and Analytical Approaches to Optimal Control Strategies in Predator-Prey Models

Abstract

Since the late 1950s, Optimal Control Theory (OCT) has been used in a variety of modelling and control engineering applications. It is also widely used in many other areas. OCT can be classified into two categories: fixed-end point-free time and free-end fixed time. We study the free-end fixed-time problem in this work. One of the most significant and fascinating areas of ecology is the study of the predator-prey system. Therefore, it's important to maximize the overall population in a predator-prey system. A large-time optimal control problem has been solved using the iterative numerical approach in Aitziber Ibanez's research. In our study, we examined the optimal mixing time needed to maximize the total population of predators and prey using the Lotka-Volterra model. To achieve this, we utilized OCT and identified the structure of the control that maximizes the total population. By analyzing the structure of control for different predation rates of prey and reproduction rates of predator values, in three cases, we were able to maximize the population at a preferred time by using the Pontryagin Maximum Principle. We partially segregate the prey population and mix after a specific time. The mixing rate is zero in two cases. This implies that it is necessary to keep separating the two populations all the time. However, this is not practical in most situations. For the third case, there is a maximum mixing time that can be found by numerical imulation or by using our estimate. Furthermore, we used MATLAB simulations to validate the results of our research. In future work, we will find out the number of switching times to simplify a complex system. This mathematical research would help the fishing industry plan their harvesting policy in a given time interval and will be important for the ecosystem to grow continually.