An Optimal Control Study with Quantity of Additional food as Control in Prey-Predator Systems involving Inhibitory Effect

Abstract

Additional food provided prey-predator systems have become a significant and important area of study for both theoretical and experimental ecologists. This is mainly because provision of additional food to the predator in the prey-predator systems has proven to facilitate wildlife conservation as well as reduction of pesticides in agriculture. Further, the mathematical modeling and analysis of these systems provide the eco-manager with various strategies that can be implemented on field to achieve the desired objectives. The outcomes of many theoretical and mathematical studies of such additional food systems have shown that the quality and quantity of additional food play a crucial role in driving the system to the desired state. However, one of the limitations of these studies is that they are asymptotic in nature, where the desired state is reached eventually with time. To overcome these limitations, we present a time optimal control study for an additional food provided prey-predator system involving inhibitory effect with quantity of additional food as the control parameter with the objective of reaching the desired state in finite (minimum) time. The results show that the optimal solution is a bang-bang control with a possibility of multiple switches. Numerical examples illustrate the theoretical findings. These results can be applied to both biological conservation and pest eradication.

Keywords

• prey-predator systems
• inhibitory effect
• additional food supplements
• optimal control problem
• bang-bang controls
• pontryagin’s maximum principle
• 49j30
• 49k30
• 92d25
• 92d40
• 92d45