Influence of quantity of additional food in achieving biological conservation and pest management in minimum-time for prey-predator systems involving Holling type III response

Incorporating additional food supplements into the predators' diet complementary to the target prey has gained importance over the years due to its pertinence in achieving biological conservation and biological control. Studies by theoretical ecologists and mathematicians reveal that by providing appropriate quality and quantity of additional food to the predator, the system could be driven either towards co-existence of species (to an admissible interior equilibrium), thereby achieving conservation or towards elimination of either of species achieving bio-control eventually with time. However, one of the limitations of these studies is that the desired state is reached only as asymptotes which makes the outcomes of the studies not that practically viable. In this work, to overcome the limitation of asymptotes, we formulate and study a time optimal control problem for additional food provided system involving type III response using quantity of additional food as the control. The objective of the study is to reach the desired terminal state in minimum time. To that end, we first prove the existence of optimal solution using the Filippov's existence theorem and then establish the characteristics of the optimal control using the Pontryagin's Maximum Principle. Using the Hamiltonian minimization condition and the monotonicity property of the Hamiltonian with respect to the quantity parameter, we show that the optimal control strategy is of bang-bang type with a possibility of multiple switches in the trajectory in case of biological conservation and no switch in case of pest management. Since the additional food system exhibits contrasting behaviour with respect to quality additional food, we have considered multiple cases of quality as a part of this study and in each case, we fixed the quality parameter as constant. The theoretical results have been illustrated by performing numerical simulations for various cases relating to both biological conservation and pest management. The theoretical outcomes of this study are in line with ecological field observations.