# American Institute of Mathematical Sciences

December  2019, 9(4): 607-621. doi: 10.3934/mcrf.2019043

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

 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

Received  May 2018 Revised  August 2019 Published  November 2019

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.

Citation: Sebastian Aniţa, Ana-Maria Moşsneagu. Optimal harvesting for age-structured population dynamics with size-dependent control. Mathematical Control & Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043
##### References:

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##### References:
The size of an individual as a function of age $a$
$\alpha$ as a function of time t. The hashed region is the area where the control acts
The fertility and mortality rates
The representation of $J$ as a function of iteration
The harvesting effort for Test 1
The harvesting effort for Test 2
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
 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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