Abstract
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.
Highlights
Prey-predator systems where predators are provided with additional food supplements are being extensively studied by biologists, ecologists as well as mathematicians [16, 31, 33, 36, 46, 48]
Motivated by the above discussion, in this article, we study the role of quantity of additional food ξ in in uencing the global dynamics of the additional food provided system (1.3) - (1.4) in achieving the desired state in minimum time
The role of quality and quality of additional food provided to predators and its impact on the ecosystem has been discussed in various works [2, 30, 31, 33, 48]
Summary
Prey-predator systems where predators are provided with additional food supplements are being extensively studied by biologists, ecologists (both theoretical and experimental ecologists) as well as mathematicians [16, 31, 33, 36, 46, 48]. Using equation (4.9), the condition (4.14), and the monotonicity property of Hamiltonian function with respect to ξ , we conclude that the optimal control solution ξ *(t) could be of bang-bang type if no singularity exists in any sub-interval of , T. This implies that optimal control function would assume the form: ξmax , if ξ. The phase space can be divided into various regions as follows: 1. Case I: F(x) has no positive roots: In this case, we can divide the phase space into two regions as follows: Region Ia :=
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