Z-type control of populations for Lotka–Volterra model with exponential convergence

• Applications of Z-type dynamic method for solving population control problem of Lotka–Volterra model are illustrated.
• Three Z-type control laws are developed.
• Control laws keep species away from extinction and improve stability of ecosystems.
• Three control laws possess exponential convergence performances.
• Theoretical analyses and numerical simulations substantiate effectiveness of such control laws.

The population control of the Lotka–Volterra model is one of the most important and widely investigated issues in mathematical ecology. In this study, assuming that birth rate is controllable and using the Z-type dynamic method, we develop Z-type control laws to drive the prey population and/or predator population to a desired state to keep species away from extinction and to improve ecosystem stability. A direct controller group is initially designed to control the prey and predator populations simultaneously. Two indirect controllers are then proposed for prey population control and predator population control by exerting exogenous measure on another species. All three control laws possess exponential convergence performances. Finally, the corresponding numerical simulations are performed. Results substantiate the theoretical analysis and effectiveness of such Z-type control laws for the population control of the Lotka–Volterra model.