Discrete-Time Optimal Control of Species Augmentation for Predator-Prey Model

Abstract

Species augmentation is one of the methods used to promote biodiversity and prevent endangered species loss and extinction. The current work applies discrete-time optimal control theory to two models of species augmentation for predator-prey relationships. In discrete-time models, the order in which events occur can give different qualitative results. Two models representing different orders of events of optimal augmentation timing are considered. In one model, the population grows and predator-prey action occurs before the translocation of reserve species for augmentation. In the second model, the augmentation happens first and is followed by growth and then predator-prey action. The reserve and target populations are subjected to strong Allee effects. The optimal augmentation models employed in this work aim to maximize the prey (target population) and reserve population at the final time and minimize the associated cost at each time step. Numerical simulations in the two models are conducted using the discrete version of the forward-backward sweep method and the sequential quadratic programming iterative method, respectively. The simulation results show different population levels in the two models under varying parameter scenarios. Objective functional values showing percentage increases with optimal controls are calculated for each simulation. Different optimal augmentation strategies for the two orders of events are discussed. This work represents the first optimal augmentation results for models incorporating the predator-prey relationship with discrete events.