# 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 and Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043
##### References:
 [1] B. Ainseba, S. Aniţa and M. Langlais, Optimal control for a nonlinear age-structured population dynamics model, Electron. J. Diff. Eqs., 28 (2003), 1-9. [2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Acad. Publ., Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [3] L.-I. Aniţa, S. Aniţa and V. Arnǎutu, Optimal harvesting for periodic age-dependent population dynamics with logistic term, Appl. Math. Comput., 215 (2009), 2701-2715.  doi: 10.1016/j.amc.2009.09.010. [4] S. Aniţa, V. Arnǎutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics. From Mathematical Models to Numerical Simulation with MATLAB, Birkhäuser, Basel, 2011. doi: 10.1007/978-0-8176-8098-5. [5] S. Aniţa, V. Capasso and A.-M. Moşneagu, Regional control in optimal harvesting problems of population dynamics, Nonlin. Anal., 147 (2016), 191-212.  doi: 10.1016/j.na.2016.09.008. [6] V. Arnǎutu and P. Neittaanmäki, Optimal Control from Theory to Computer Programs, Kluwer Acad. Publ., Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3. [7] A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phen., 9 (2014), 20-37.  doi: 10.1051/mmnp/20149403. [8] A. O. Belyakov and V. M. Veliov, On optimal harvesting in age-structured populations, in Dynamic Perspectives on Managerial Decision Making (H. Dawid, K.F. Doerner, G. Feichtinger, P.M. Kort, A. Seidl, Eds.), Springer Internat. Publ., 22 (2016), 149-166. doi: 10.1007/978-3-319-39120-5_9. [9] A. Bressan, G. M. Coclite and W. Shen, A multidimensional optimal-harvesting problem with measure-valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202.  doi: 10.1137/110853510. [10] M. Brokate, Pontyagin's principle for control problems in age-dependent population dynamics, J. Math. Biol., 23 (1985), 75-101.  doi: 10.1007/BF00276559. [11] M. Brokate, On a certain optimal harvesting problem with continuous age structure, in: Optimal Control of Partial Differential Equations II (K.-H. Hoffmann, W. Krabs, Eds.), Birkhäuser, Boston, 78 (1987), 29-42. [12] G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 55 (2017), 913-935.  doi: 10.1137/16M1061886. [13] G. M. Coclite, M. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 865-900.  doi: 10.3934/dcdss.2018053. [14] G. Feichtinger, G. Tragler and V. M. Veliov, Optimality conditions for age-structured control systems, J. Math. Anal. Appl., 288 (2003), 47-68.  doi: 10.1016/j.jmaa.2003.07.001. [15] K. R. Fister and S. Lenhart, Optimal harvesting in an age-structured predator-prey model, Appl. Math. Optim., 54 (2006), 1-15.  doi: 10.1007/s00245-005-0847-9. [16] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of age-structured populations: some simple models, Math. Biosci., 55 (1981), 115-136.  doi: 10.1016/0025-5564(81)90015-8. [17] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of persistent age-structured populations, J. Math. Biol., 13 (1981), 131-148.  doi: 10.1007/BF00275209. [18] Z. R. He, Optimal harvesting of two competing species with age dependence, Nonlin. Anal. Real World Appl., 7 (2006), 769-788.  doi: 10.1016/j.nonrwa.2005.04.005. [19] N. Hegoburu, P. Magal and M. Tucsnak, Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750, https://hal.archives-ouvertes.fr/hal-01395712. doi: 10.1137/16M1103087. [20] N. Hritonenko and Y. Yatsenko, Optimization of harvesting age in integral age-dependent model of population dynamics, Math. Biosci., 195 (2005), 154-167.  doi: 10.1016/j.mbs.2005.03.001. [21] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs - C.N.R., Giardini Editori e Stampatori, Pisa, 1995. [22] S. Lenhart, Using optimal control of parabolic PDEs to investigate population questions, NIMBioS, April 9-11, 2014, https://www.fields.utoronto.ca/programs/scientific/13-14/BIOMAT/presentations/lenhartToronto3.pdf. [23] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, 2007. [24] Z. Luo, Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model, Appl. Math. Comput., 186 (2007), 1742-1752.  doi: 10.1016/j.amc.2006.08.168. [25] Z. Luo, W. T. Li and M. Wang, Optimal harvesting control problem for linear periodic age-dependent population dynamics, Appl. Math. Comput., 151 (2004), 789-800.  doi: 10.1016/S0096-3003(03)00536-8. [26] L. F. Murphy and S. J. Smith, Optimal harvesting of an age-structured population, J. Math. Biol., 29 (1990), 77-90.  doi: 10.1007/BF00173910. [27] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. [28] C. Zhao, M. Wang and P. Zhao, Optimal harvesting problems for age-dependent interacting species with diffusion, Appl. Math. Comput., 163 (2005), 117-129.  doi: 10.1016/j.amc.2004.01.030. [29] C. Zhao, P. Zhao and M. Wang, Optimal harvesting for nonlinear age-dependent population dynamics, Math. Comput. Model., 43 (2006), 310-319.  doi: 10.1016/j.mcm.2005.06.008.

show all references

##### References:
 [1] B. Ainseba, S. Aniţa and M. Langlais, Optimal control for a nonlinear age-structured population dynamics model, Electron. J. Diff. Eqs., 28 (2003), 1-9. [2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Acad. Publ., Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [3] L.-I. Aniţa, S. Aniţa and V. Arnǎutu, Optimal harvesting for periodic age-dependent population dynamics with logistic term, Appl. Math. Comput., 215 (2009), 2701-2715.  doi: 10.1016/j.amc.2009.09.010. [4] S. Aniţa, V. Arnǎutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics. From Mathematical Models to Numerical Simulation with MATLAB, Birkhäuser, Basel, 2011. doi: 10.1007/978-0-8176-8098-5. [5] S. Aniţa, V. Capasso and A.-M. Moşneagu, Regional control in optimal harvesting problems of population dynamics, Nonlin. Anal., 147 (2016), 191-212.  doi: 10.1016/j.na.2016.09.008. [6] V. Arnǎutu and P. Neittaanmäki, Optimal Control from Theory to Computer Programs, Kluwer Acad. Publ., Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3. [7] A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phen., 9 (2014), 20-37.  doi: 10.1051/mmnp/20149403. [8] A. O. Belyakov and V. M. Veliov, On optimal harvesting in age-structured populations, in Dynamic Perspectives on Managerial Decision Making (H. Dawid, K.F. Doerner, G. Feichtinger, P.M. Kort, A. Seidl, Eds.), Springer Internat. Publ., 22 (2016), 149-166. doi: 10.1007/978-3-319-39120-5_9. [9] A. Bressan, G. M. Coclite and W. Shen, A multidimensional optimal-harvesting problem with measure-valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202.  doi: 10.1137/110853510. [10] M. Brokate, Pontyagin's principle for control problems in age-dependent population dynamics, J. Math. Biol., 23 (1985), 75-101.  doi: 10.1007/BF00276559. [11] M. Brokate, On a certain optimal harvesting problem with continuous age structure, in: Optimal Control of Partial Differential Equations II (K.-H. Hoffmann, W. Krabs, Eds.), Birkhäuser, Boston, 78 (1987), 29-42. [12] G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 55 (2017), 913-935.  doi: 10.1137/16M1061886. [13] G. M. Coclite, M. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 865-900.  doi: 10.3934/dcdss.2018053. [14] G. Feichtinger, G. Tragler and V. M. Veliov, Optimality conditions for age-structured control systems, J. Math. Anal. Appl., 288 (2003), 47-68.  doi: 10.1016/j.jmaa.2003.07.001. [15] K. R. Fister and S. Lenhart, Optimal harvesting in an age-structured predator-prey model, Appl. Math. Optim., 54 (2006), 1-15.  doi: 10.1007/s00245-005-0847-9. [16] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of age-structured populations: some simple models, Math. Biosci., 55 (1981), 115-136.  doi: 10.1016/0025-5564(81)90015-8. [17] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of persistent age-structured populations, J. Math. Biol., 13 (1981), 131-148.  doi: 10.1007/BF00275209. [18] Z. R. He, Optimal harvesting of two competing species with age dependence, Nonlin. Anal. Real World Appl., 7 (2006), 769-788.  doi: 10.1016/j.nonrwa.2005.04.005. [19] N. Hegoburu, P. Magal and M. Tucsnak, Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750, https://hal.archives-ouvertes.fr/hal-01395712. doi: 10.1137/16M1103087. [20] N. Hritonenko and Y. Yatsenko, Optimization of harvesting age in integral age-dependent model of population dynamics, Math. Biosci., 195 (2005), 154-167.  doi: 10.1016/j.mbs.2005.03.001. [21] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs - C.N.R., Giardini Editori e Stampatori, Pisa, 1995. [22] S. Lenhart, Using optimal control of parabolic PDEs to investigate population questions, NIMBioS, April 9-11, 2014, https://www.fields.utoronto.ca/programs/scientific/13-14/BIOMAT/presentations/lenhartToronto3.pdf. [23] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, 2007. [24] Z. Luo, Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model, Appl. Math. Comput., 186 (2007), 1742-1752.  doi: 10.1016/j.amc.2006.08.168. [25] Z. Luo, W. T. Li and M. Wang, Optimal harvesting control problem for linear periodic age-dependent population dynamics, Appl. Math. Comput., 151 (2004), 789-800.  doi: 10.1016/S0096-3003(03)00536-8. [26] L. F. Murphy and S. J. Smith, Optimal harvesting of an age-structured population, J. Math. Biol., 29 (1990), 77-90.  doi: 10.1007/BF00173910. [27] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. [28] C. Zhao, M. Wang and P. Zhao, Optimal harvesting problems for age-dependent interacting species with diffusion, Appl. Math. Comput., 163 (2005), 117-129.  doi: 10.1016/j.amc.2004.01.030. [29] C. Zhao, P. Zhao and M. Wang, Optimal harvesting for nonlinear age-dependent population dynamics, Math. Comput. Model., 43 (2006), 310-319.  doi: 10.1016/j.mcm.2005.06.008.
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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