Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam, Puttaparthi, Anantapur District, 515134 Andhra Pradesh, India.
Heliyon. 2021 Aug 3;7(8):e07699. doi: 10.1016/j.heliyon.2021.e07699. eCollection 2021 Aug.
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 and then establish the characteristics of the optimal control using the . 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.
多年来，在捕食者的饮食中加入与目标猎物互补的额外食物补充剂变得越来越重要，因为这对于实现生物保护和生物控制具有相关性。理论生态学家和数学家的研究表明，通过向捕食者提供质量和数量合适的额外食物，系统可以朝着物种共存（达到可接受的内部平衡）发展，从而实现保护，或者随着时间的推移朝着消除其中一个物种以实现生物控制发展。然而，这些研究的局限性之一是，期望状态只能作为渐近线达到，这使得研究结果在实际中不太可行。在这项工作中，为了克服渐近线的局限性，我们针对使用额外食物数量作为控制量的包含III型响应的额外食物提供系统，制定并研究了一个时间最优控制问题。该研究的目标是在最短时间内达到期望的终端状态。为此，我们首先使用[具体方法1]证明最优解的存在性，然后使用[具体方法2]建立最优控制的特性。利用哈密顿量最小化条件以及哈密顿量关于数量参数的单调性，我们表明最优控制策略是bang-bang型的，在生物保护情况下轨迹可能有多个切换，而在害虫管理情况下没有切换。由于额外食物系统在额外食物质量方面表现出不同的行为，我们在本研究中考虑了多种质量情况，并且在每种情况下，我们将质量参数固定为常数。通过对与生物保护和害虫管理相关的各种情况进行数值模拟，对理论结果进行了说明。本研究的理论结果与生态实地观察结果一致。
