Time optimal control of an additional food provided predator–prey system with applications to pest management and biological conservation

Abstract

Use of additional food has been widely recognized by experimental scientists as one of the important tools for biological control such as species conservation and pest management. The quality and quantity of additional food supplied to the predators is known to play a vital role in the controllability of the system. The present study is continuation of a previous work that highlights the importance of quality and quantity of the additional food in the dynamics of a predator-prey system in the context of biological control. In this article the controllability of the predator-prey system is analyzed by considering inverse of quality of the additional food as the control variable. Control strategies are offered to steer the system from a given initial state to a required terminal state in a minimum time by formulating Mayer problem of optimal control. It is observed that an optimal strategy is a combination of bang-bang controls and could involve multiple switches. Properties of optimal paths are derived using necessary conditions for Mayer problem. In the light of the results evolved in this work it is possible to eradicate the prey from the eco-system in the minimum time by providing the predator with high quality additional food, which is relevant in the pest management. In the perspective of biological conservation this study highlights the possibilities to drive the state to an admissible interior equilibrium (irrespective of its stability nature) of the system in a minimum time.