The switch point algorithm applied to a harvesting problem

Summer Atkins; William W. Hager; Maia Martcheva; Summer Atkins; William W. Hager; Maia Martcheva

doi:10.3934/mbe.2024269

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 5: 6123-6149. doi: 10.3934/mbe.2024269

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The switch point algorithm applied to a harvesting problem

1.
Department of Mathematical Sciences, University of Alabama in Huntsville, AL 35899, USA
2.
Department of Mathematics, University of Florida, FL 32601, USA

Received: 17 January 2024 Revised: 24 March 2024 Accepted: 06 May 2024 Published: 30 May 2024

In this paper, we investigate an optimal harvesting problem of a spatially explicit fishery model that was previously analyzed. On the surface, this problem looks innocent, but if parameters are set to where a singular arc occurs, two complex questions arise. The first question pertains to Fuller's phenomenon (or chattering), a phenomenon in which the optimal control possesses a singular arc that cannot be concatenated with the bang-bang arcs without prompting infinite oscillations over a finite region. 1) How do we numerically assess whether or not a problem chatters in cases when we cannot analytically prove such a phenomenon? The second question focuses on implementation of an optimal control. 2) When an optimal control has regions that are difficult to implement, how can we find alternative strategies that are both suboptimal and realistic to use? Although the former question does not apply to all optimal harvesting problems, most fishery managers should be concerned about the latter. Interestingly, for this specific problem, our techniques for answering the first question results in an answer to the the second. Our methods involve using an extended version of the switch point algorithm (SPA), which handles control problems having initial and terminal conditions on the states. In our numerical experiments, we obtain strong empirical evidence that the harvesting problem chatters, and we find three alternative harvesting strategies with fewer switches that are realistic to implement and near optimal.
- fish harvesting,
- singular control,
- chattering,
- switch point algorithm,
- marine reserves,
- fisheries management
Citation: Summer Atkins, William W. Hager, Maia Martcheva. The switch point algorithm applied to a harvesting problem[J]. Mathematical Biosciences and Engineering, 2024, 21(5): 6123-6149. doi: 10.3934/mbe.2024269

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Abstract

In this paper, we investigate an optimal harvesting problem of a spatially explicit fishery model that was previously analyzed. On the surface, this problem looks innocent, but if parameters are set to where a singular arc occurs, two complex questions arise. The first question pertains to Fuller's phenomenon (or chattering), a phenomenon in which the optimal control possesses a singular arc that cannot be concatenated with the bang-bang arcs without prompting infinite oscillations over a finite region. 1) How do we numerically assess whether or not a problem chatters in cases when we cannot analytically prove such a phenomenon? The second question focuses on implementation of an optimal control. 2) When an optimal control has regions that are difficult to implement, how can we find alternative strategies that are both suboptimal and realistic to use? Although the former question does not apply to all optimal harvesting problems, most fishery managers should be concerned about the latter. Interestingly, for this specific problem, our techniques for answering the first question results in an answer to the the second. Our methods involve using an extended version of the switch point algorithm (SPA), which handles control problems having initial and terminal conditions on the states. In our numerical experiments, we obtain strong empirical evidence that the harvesting problem chatters, and we find three alternative harvesting strategies with fewer switches that are realistic to implement and near optimal.

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