Optimal control harvesting in a deterministic predator-prey model for a three-patch ecosystem

Lucian Talu Mayabi; David Angwenyi; Duncan Oganga

doi:10.51867/Asarev.Maths.2.1.6

Authors

Lucian Talu Mayabi Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya Author https://orcid.org/0000-0002-4546-164X
David Angwenyi Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya Author https://orcid.org/0000-0002-6958-2817
Duncan Oganga Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. Box 190-50100, Kakamega, Kenya Author https://orcid.org/0000-0003-3727-9853

DOI:

https://doi.org/10.51867/Asarev.Maths.2.1.6

Abstract

Predator-prey interactions play a pivotal role in shaping ecosystem dynamics, with significant implications for sustainable resource management. While much of the existing literature has focused on deterministic harvesting models in single or two-patch ecosystems, there remains a gap in exploring optimal harvesting strategies within spatially distributed multi-patch systems. This study develops a deterministic predator-prey model across three interconnected patches, representing cage-based aquaculture zones within a lake. The model incorporates prey migration between patches and harvesting applied to the prey population only. We formulate an optimal control problem aimed at maximizing the net economic return from harvesting while ensuring ecological sustainability. Using Pontryagin’s Maximum Principle, we derive necessary conditions for optimality and construct a time-dependent control strategy. Numerical simulations, with the help of Python software, reveal that fixed-rate harvesting lead to population decline or instability, while the optimal control strategy stabilizes both predator and prey populations within sustainable bounds. These findings provide valuable insights for fisheries management, aquaculture policy, and the design of ecologically sound harvesting strategies in multi-zone environments.

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