Optimal harvesting for age-structured population dynamics with size-dependent control

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doi:10.3934/mcrf.2019043

Article Contents

2019, Volume 9, Issue 4: 607-621. Doi: 10.3934/mcrf.2019043

This issue Previous Article Preface to the special issue on control of infinite dimensional systems Next Article Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Optimal harvesting for age-structured population dynamics with size-dependent control

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1.
Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, "Octav Mayer" Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania
2.
Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, Iaşi 700506, Romania

^* Corresponding author: Sebastian Aniţta
^* Corresponding author: Sebastian Aniţta

Received: April 30, 2018

Revised: July 31, 2019

Published: November 2019

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Abstract

We investigate two optimal harvesting problems related to age-dependent population dynamics; namely we consider two problems of maximizing the profit for age-structured population dynamics with respect to a size-dependent harvesting effort. We evaluate the directional derivatives for the cost functionals. The structure of the harvesting effort is uniquely determined by its intensity (magnitude) and by its area of action/distribution. We derive an iterative algorithm to increase at each iteration the profit by changing the intensity of the harvesting effort and its distribution area. Some numerical tests are given to illustrate the effectiveness of the theoretical results for the first optimal harvesting problem.

Keywords:

Age-structured population dynamics,
optimal harvesting problem; optimal control,
harvesting effort,
size-dependence,
optimality conditions,
iterative algorithm.

Mathematics Subject Classification: Primary: 92D25, 49K20; Secondary: 35Q92, 35Q93, 49M05.

Citation:

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Figure 1. The size of an individual as a function of age $ a $

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Figure 2. $ \alpha $ as a function of time t. The hashed region is the area where the control acts

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Figure 3. The fertility and mortality rates

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Figure 4. The representation of $ J $ as a function of iteration

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Figure 5. The harvesting effort for Test 1

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Figure 6. The harvesting effort for Test 2

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Table 1. The value of $ J $ at each iteration

iteration	J
1	0.464525126289841
2	0.533792098410522
3	0.545212800519842
4	0.552867826650825
5	0.556828710910306
6	0.558793583997659
7	0.559787107591890
8	0.560285396165302
9	0.560534749191512
10	0.560659455111737
11	0.560721812501533
12	0.560752991933830
13	0.560768581787776
14	0.560776376743360

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