Optimal Control Strategies for Predator–Prey Systems with Bang–Bang and Quadratic Control Terms.

Optimal control theory provides a powerful mathematical framework for formulating effective ecological management strategies, particularly for complex systems such as predator-prey interactions. While foundational models like the Lotka-Volterra equations offer important insights into species relationships, their use in sustainable management is limited by simplifying assumptions, particularly the omission of intraspecific competition. Modern research has addressed these limitations by integrating more realistic ecological mechanisms and applying optimal control theory as a powerful mathematical method. This study extends that approach by applying optimal control theory on a predator-prey model that includes internal competition in both species, investigating two distinct management goals. The first strategy seeks to conserve predators through prey addition, where controlled prey supplementation supports predator growth in finite time while maintaining prey availability and overall ecosystem stability. The objective is to increase the predator population while quadratically penalizing control effort through a running cost functional. The second strategy aims to maximize the total population by controlling the interaction rate between species. For this approach, a bang–bang control problem is formulated in which a linear control variable governs the mixing (or segregation) of populations, with the goal of maximizing the total population at a fixed final time. Pontryagin’s Maximum Principle is applied to derive necessary optimality conditions for both problems. Numerical simulations show that the first optimal control strategy increases predator population density in finite time through sustained prey supplementation while maintaining sufficient prey availability. For the second strategy, numerical optimization identifies a specific switching time that achieves the global maximum for the total population at the terminal time. These findings provide a theoretical foundation applicable to wildlife conservation and fishery management.