Abstract
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.
Highlights
From the phase space studies and stability analysis of the additional food system as in Srinivasu et al (2018), we see that the eventual state and stability of the system can be determined based on the values assumed by the two exogenous parameters α and κ with respect to the following curves given below under each case of initial system presented in Table 2: Prey Elimination Curve (PEC), βκ − δ(1 + ακ) = 0
Using the expression for Hamiltonian (4.7), using the Hamiltonian minimization condition (4.13), and the monotonicity property of the Hamiltonian function with respect to κ, we can conclude that the optimal control might be of bang-bang type provided no singular solution exists for a sub-interval in [0, T ]
In order to achieve co-existence of species leading to biological conservation, the optimal control function could involve multiple switching where as to achieve pest management, it is enough to provide the predators with maximum quantity of high quality additional food
Summary
Study of ecosystems where the predator is provided with alternate sources of food in addition to the target prey has gained prominence over the years and has become one of the important areas of research for biologists, theoretical and experimental ecologists, mathematicians and statisticians Harwood et al (2004); Liu et al (2018); Redpath et al (2001); Sabelis et al (2006); Sahoo and Poria (2014); Sen et al (2015); Soltaniyan et al (2020); Van Baalen et al (2001); Wade et al (2008). Some of the outcomes of the mathematical studies of additional food systems Das and Samanta (2018a,b, 2020); Liu et al (2018); Mondal and Samanta (2019, 2020); Prasad and Prasad (2019); Sahoo and Poria (2014); Sen et al (2015); Srinivasu et al (2007, 2018); Vamsi et al (2019) reveal that the provision of additional food to the predators affects the global dynamics of the system causing indirect interaction of species. Motivated by the aforementioned studies, in this article, we investigate the role of quantity of additional food κ in the global dynamics of the additional food provided system (1.5) - (1.6) in achieving the desired state in minimum (finite) time This is done as follows: we determine the admissible states that could be reached by varying the quantity in the range [κmin, κmax].
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