# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021245
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## Maximum principle for the optimal harvesting problem of a size-stage-structured population model

 1 College of science, Qingdao University of Technology, Qingdao, 266033, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

*Corresponding author: Miaomiao Chen

Received  March 2021 Revised  August 2021 Early access October 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (Nos. 11771044, 11871007 and 12171039)

The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.

Citation: Miaomiao Chen, Rong Yuan. Maximum principle for the optimal harvesting problem of a size-stage-structured population model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021245
##### References:
 [1] B. Ainseba, Z. Feng, M. Iannelli and F. A. Milner, Control strategies for TB epidemics, SIAM J. Appl. Math., 77 (2017), 82-107.  doi: 10.1137/15M1048719. [2] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177.  doi: 10.1007/s002850050124. [3] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.  doi: 10.1126/science.197.4302.463. [4] K. Belkhodja, A. Moussaoui and M. A. Aziz Alaoui, Optimal harvesting and stability for a prey-predator model, Nonlinear Anal. Real World Appl., 39 (2018), 321-336.  doi: 10.1016/j.nonrwa.2017.07.004. [5] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of persistent age-structured populations, J. Math. Biol., 13 (1981/82), 131-148.  doi: 10.1007/BF00275209. [6] J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., 1980. [7] A. Ishaku, A. M. Gazali, S. A. Abdullahi and N. Hussaini, Analysis and optimal control of an HIV model based on CD4 count, J. Math. Biol., 81 (2020), 209-241.  doi: 10.1007/s00285-020-01508-8. [8] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482.  doi: 10.3934/dcdsb.2002.2.473. [9] N. Kato, Optimal harvesting for nonlinear size-structured population dynamics, J. Math. Anal. Appl., 342 (2008), 1388-13988.  doi: 10.1016/j.jmaa.2008.01.010. [10] N. Kato, Linear size-structured population models with spacial diffusion and optimal harvesting problems, Math. Model. Nat. Phenom., 9 (2014), 122-130.  doi: 10.1051/mmnp/20149408. [11] M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551-577.  doi: 10.1137/15M1021271. [12] H. D. Kwon, J. Lee and M. Yoon, An age-structured model with immune response of HIV infection: Modeling and optimal control approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 153-172.  doi: 10.3934/dcdsb.2014.19.153. [13] Y. Li, Z. Zhang, Y. Lv and Z. Liu, Optimal harvesting for a size-stage-structured population model, Nonlinear Anal. Real World Appl., 44 (2018), 616-630.  doi: 10.1016/j.nonrwa.2018.06.001. [14] M. Liu and C. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1493-1508.  doi: 10.3934/dcdsb.2017071. [15] P. Magal and Z. Zhang, Competition for light in forest population dynamics: From computer simulator to mathematical model, J. Theoret. Biol., 419 (2017), 290-304.  doi: 10.1016/j.jtbi.2017.02.025. [16] P. Magal and Z. Zhang, A system of state-dependent delay differential equation modelling forest growth II: Boundedness of solutions, Nonlinear Anal. Real World Appl., 42 (2018), 334-352.  doi: 10.1016/j.nonrwa.2018.01.002. [17] T. McMahon, Size and shape in biology: Elastic criteria impose limits on biological proportions, and consequently on metabolic rates, Science, 179 (1973), 1201-1204. [18] X. Meng, Niklas L. P. Lundström, M. Bodin and Å ke Brännström, Dynamics and management of stage-structured fish stocks, Bull. Math. Biol., 75 (2013), 1-23.  doi: 10.1007/s11538-012-9789-y. [19] J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 1986. [20] R. M. Nisbet and W. S. C. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoret. Population Biol., 23 (1983), 114-135.  doi: 10.1016/0040-5809(83)90008-4. [21] Y. Pei, M. Chen, X. Liang and C. Li, Model-based on fishery management systems with selective harvest policies, Math. Comput. Simulation, 156 (2019), 377-395.  doi: 10.1016/j.matcom.2018.08.009. [22] H. L. Smith, Reduction of structured population models to threshold-type delay equations and functional-differential equations: A case study, Math. Biosci., 113 (1993), 1-23.  doi: 10.1016/0025-5564(93)90006-V. [23] H. L. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mountain J. Math., 24 (1994), 311-334.  doi: 10.1216/rmjm/1181072468. [24] H. L. Smith, A structured population model and a related functional differential equation: Global attractors and uniform persistence, J. Dynam. Differential Equations, 6 (1994), 71-99.  doi: 10.1007/BF02219189. [25] H. L. Smith, Equivalent dynamics for a structured population model and a related functional-differential equation, Rocky Mountain J. Math., 25 (1995), 491-499.  doi: 10.1216/rmjm/1181072298. [26] X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170 (2001), 173-186.  doi: 10.1016/S0025-5564(00)00068-7. [27] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, NJ, 2003. [28] P. Waltman, Deterministic Threshold Models in the Theory of Epidemics, Springer-Verlag, Berlin-New York, 1974. [29] Z. Wang, Optimal harvesting for age distribution and weighted size competitive species with diffusion, J. Comput. Appl. Math., 328 (2018), 485-496.  doi: 10.1016/j.cam.2017.07.026. [30] E. E. Werner and J. F. Gilliam, The ontogenetic niche and species interactions in size-structured populations, Annu. Rev. Ecol. Syst., 15 (1984), 393-425.  doi: 10.1146/annurev.es.15.110184.002141. [31] A. Zenati, M. Chakir and M. Tadjine, Global stability analysis and optimal control therapy of blood cell production process (hematopoiesis) in acute myeloid leukemia, J. Theoret. Biol., 458 (2018), 15-30.  doi: 10.1016/j.jtbi.2018.09.001. [32] F. Zhang, R. Liu and Y. Chen, Optimal harvesting in a periodic food chain model with size structures in predators, Appl. Math. Optim., 75 (2017), 229-251.  doi: 10.1007/s00245-016-9331-y. [33] X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. Real World Appl., 4 (2003), 639-651.  doi: 10.1016/S1468-1218(02)00084-6.

show all references

##### References:
 [1] B. Ainseba, Z. Feng, M. Iannelli and F. A. Milner, Control strategies for TB epidemics, SIAM J. Appl. Math., 77 (2017), 82-107.  doi: 10.1137/15M1048719. [2] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177.  doi: 10.1007/s002850050124. [3] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.  doi: 10.1126/science.197.4302.463. [4] K. Belkhodja, A. Moussaoui and M. A. Aziz Alaoui, Optimal harvesting and stability for a prey-predator model, Nonlinear Anal. Real World Appl., 39 (2018), 321-336.  doi: 10.1016/j.nonrwa.2017.07.004. [5] M. E. Gurtin and L. F. Murphy, On the optimal harvesting of persistent age-structured populations, J. Math. Biol., 13 (1981/82), 131-148.  doi: 10.1007/BF00275209. [6] J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., 1980. [7] A. Ishaku, A. M. Gazali, S. A. Abdullahi and N. Hussaini, Analysis and optimal control of an HIV model based on CD4 count, J. Math. Biol., 81 (2020), 209-241.  doi: 10.1007/s00285-020-01508-8. [8] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482.  doi: 10.3934/dcdsb.2002.2.473. [9] N. Kato, Optimal harvesting for nonlinear size-structured population dynamics, J. Math. Anal. Appl., 342 (2008), 1388-13988.  doi: 10.1016/j.jmaa.2008.01.010. [10] N. Kato, Linear size-structured population models with spacial diffusion and optimal harvesting problems, Math. Model. Nat. Phenom., 9 (2014), 122-130.  doi: 10.1051/mmnp/20149408. [11] M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551-577.  doi: 10.1137/15M1021271. [12] H. D. Kwon, J. Lee and M. Yoon, An age-structured model with immune response of HIV infection: Modeling and optimal control approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 153-172.  doi: 10.3934/dcdsb.2014.19.153. [13] Y. Li, Z. Zhang, Y. Lv and Z. Liu, Optimal harvesting for a size-stage-structured population model, Nonlinear Anal. Real World Appl., 44 (2018), 616-630.  doi: 10.1016/j.nonrwa.2018.06.001. [14] M. Liu and C. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1493-1508.  doi: 10.3934/dcdsb.2017071. [15] P. Magal and Z. Zhang, Competition for light in forest population dynamics: From computer simulator to mathematical model, J. Theoret. Biol., 419 (2017), 290-304.  doi: 10.1016/j.jtbi.2017.02.025. [16] P. Magal and Z. Zhang, A system of state-dependent delay differential equation modelling forest growth II: Boundedness of solutions, Nonlinear Anal. Real World Appl., 42 (2018), 334-352.  doi: 10.1016/j.nonrwa.2018.01.002. [17] T. McMahon, Size and shape in biology: Elastic criteria impose limits on biological proportions, and consequently on metabolic rates, Science, 179 (1973), 1201-1204. [18] X. Meng, Niklas L. P. Lundström, M. Bodin and Å ke Brännström, Dynamics and management of stage-structured fish stocks, Bull. Math. Biol., 75 (2013), 1-23.  doi: 10.1007/s11538-012-9789-y. [19] J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 1986. [20] R. M. Nisbet and W. S. C. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoret. Population Biol., 23 (1983), 114-135.  doi: 10.1016/0040-5809(83)90008-4. [21] Y. Pei, M. Chen, X. Liang and C. Li, Model-based on fishery management systems with selective harvest policies, Math. Comput. Simulation, 156 (2019), 377-395.  doi: 10.1016/j.matcom.2018.08.009. [22] H. L. Smith, Reduction of structured population models to threshold-type delay equations and functional-differential equations: A case study, Math. Biosci., 113 (1993), 1-23.  doi: 10.1016/0025-5564(93)90006-V. [23] H. L. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mountain J. Math., 24 (1994), 311-334.  doi: 10.1216/rmjm/1181072468. [24] H. L. Smith, A structured population model and a related functional differential equation: Global attractors and uniform persistence, J. Dynam. Differential Equations, 6 (1994), 71-99.  doi: 10.1007/BF02219189. [25] H. L. Smith, Equivalent dynamics for a structured population model and a related functional-differential equation, Rocky Mountain J. Math., 25 (1995), 491-499.  doi: 10.1216/rmjm/1181072298. [26] X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170 (2001), 173-186.  doi: 10.1016/S0025-5564(00)00068-7. [27] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, NJ, 2003. [28] P. Waltman, Deterministic Threshold Models in the Theory of Epidemics, Springer-Verlag, Berlin-New York, 1974. [29] Z. Wang, Optimal harvesting for age distribution and weighted size competitive species with diffusion, J. Comput. Appl. Math., 328 (2018), 485-496.  doi: 10.1016/j.cam.2017.07.026. [30] E. E. Werner and J. F. Gilliam, The ontogenetic niche and species interactions in size-structured populations, Annu. Rev. Ecol. Syst., 15 (1984), 393-425.  doi: 10.1146/annurev.es.15.110184.002141. [31] A. Zenati, M. Chakir and M. Tadjine, Global stability analysis and optimal control therapy of blood cell production process (hematopoiesis) in acute myeloid leukemia, J. Theoret. Biol., 458 (2018), 15-30.  doi: 10.1016/j.jtbi.2018.09.001. [32] F. Zhang, R. Liu and Y. Chen, Optimal harvesting in a periodic food chain model with size structures in predators, Appl. Math. Optim., 75 (2017), 229-251.  doi: 10.1007/s00245-016-9331-y. [33] X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. Real World Appl., 4 (2003), 639-651.  doi: 10.1016/S1468-1218(02)00084-6.
Time-series of (a) optimal harvesting effort, (b) adult population
Time-series of (a) growth function, (b) total juvenile population, and juvenile population distributions (c) without harvesting, (d) with optimal harvesting
(a) Time-series of the optimal harvesting effort, adult population, and total juvenile population, (b) contours of the juvenile population distribution
Time-series of (a) adult population and growth function, (b) total juvenile population and harvested adult
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