Nonlinear optimal control of population systems: applications in ecosystems

This article investigates the chaotic Lotka–Volterra system as an optimal nonlinear design problem for biological pest control strategies. In the biological control strategy, natural enemies are introduced such that the pest density is stabilized below the economic injury level, and the population of natural enemies remains sufficiently high to control the pests. Applying dynamic programming, this problem was reduced to the Hamilton–Jacobi–Bellman equation. The functions satisfying the reduced equation were obtained among the correspondent Lyapunov functions of the considered Lotka–Volterra system. A closed-form optimal feedback control law was derived. The effectiveness of the method is verified by numerical simulations of biological pest control. The biological implications of these results are also discussed.