Optimal development problem for a nonlinear population model with size structure in a periodic environment

Rong Liu; Xin Yi; Yanmei Wang; Rong Liu; Xin Yi; Yanmei Wang

doi:10.3934/math.2025573

AIMS Mathematics

2025, Volume 10, Issue 5: 12726-12744. doi: 10.3934/math.2025573

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Optimal development problem for a nonlinear population model with size structure in a periodic environment

1.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
2.
School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619, China

Received: 19 February 2025 Revised: 06 May 2025 Accepted: 21 May 2025 Published: 30 May 2025
MSC : 35F50, 49K15, 49K20, 92D25

This paper investigated the optimal development problem of a size-structured population model under a periodic environmental setting. The boundary condition of the novel model consists of a nonlinear recruitment process and a bounded input, which endow the model with more realistic and complex characteristics compared to traditional ones. First, we established the existence of a unique non-negative bounded solution and demonstrated the continuous dependence of the solutions on the control variable. Next, we showed that the adjoint system is also well-posed. Then, the Euler-Lagrange equations describing the exact structure of the optimal strategies were derived and the existence of a unique optimal policy was proved. Finally, some numerical results were presented. The obtained research results will contribute to the development of some renewable resources, such as fish resources.
- environmental periodicity,
- size structure,
- optimal harvesting,
- nonlinear recruitment process
Citation: Rong Liu, Xin Yi, Yanmei Wang. Optimal development problem for a nonlinear population model with size structure in a periodic environment[J]. AIMS Mathematics, 2025, 10(5): 12726-12744. doi: 10.3934/math.2025573

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Abstract

This paper investigated the optimal development problem of a size-structured population model under a periodic environmental setting. The boundary condition of the novel model consists of a nonlinear recruitment process and a bounded input, which endow the model with more realistic and complex characteristics compared to traditional ones. First, we established the existence of a unique non-negative bounded solution and demonstrated the continuous dependence of the solutions on the control variable. Next, we showed that the adjoint system is also well-posed. Then, the Euler-Lagrange equations describing the exact structure of the optimal strategies were derived and the existence of a unique optimal policy was proved. Finally, some numerical results were presented. The obtained research results will contribute to the development of some renewable resources, such as fish resources.

References

[1]	P. Magal, S. Ruan, Structured population models in biology and epidemiology, 1 Ed., Springer, 2008. https://doi.org/10.1007/978-3-540-78273-5
[2]	M. Iannelli, F. Milner, The basic approach to age-structured population dynamics, Springer, 2008. https://doi.org/10.1007/978-94-024-1146-1
[3]	L. Aniţa, S. Aniţa, V. Arn$\breve{a}$utu, Global behavior for an age-dependent population model with logistic term and periodic vital rates, Appl. Math. Comput., 206 (2008), 368–379. https://doi.org/10.1016/j.amc.2008.09.016 doi: 10.1016/j.amc.2008.09.016
[4]	N. Bairagi, D. Jana, Age-structured predator-prey model with habitat complexity: oscillations and control, Dyn. Syst., 27 (2012), 475–499. https://doi.org/10.1080/14689367.2012.723678 doi: 10.1080/14689367.2012.723678
[5]	B. Kakumani, S. Tumuluri, Extinction and blow-up phenomena in a non-linear gender structured population model, Nonlinear Anal.: Real World Appl., 28 (2016), 290–299. https://doi.org/10.1016/j.nonrwa.2015.10.005 doi: 10.1016/j.nonrwa.2015.10.005
[6]	E. Park, M. Iannelli, M. Kim, S. Anita, Optimal harvesting for periodic age-dependent population dynamics, SIAM J. Appl. Math., 58 (1998), 1648–1666. https://doi.org/10.1137/S0036139996301180 doi: 10.1137/S0036139996301180
[7]	B. Skritek, V. Veliov, On the infinite-horizon optimal control of age-structured systems, J. Optim. Theory Appl., 167 (2015), 243–271. https://doi.org/10.1007/s10957-014-0680-x doi: 10.1007/s10957-014-0680-x
[8]	N. Osmolovskii, V. Veliov, Optimal control of age-structured systems with mixed state-control constraints, J. Math. Anal. Appl., 455 (2017), 396–421. https://doi.org/10.1016/j.jmaa.2017.05.069 doi: 10.1016/j.jmaa.2017.05.069
[9]	Z. He, D. Ni, S. Wang, Optimal harvesting of a hierarchical age-structured population system, Int. J. Biomath., 12 (2019), 1950091. https://doi.org/10.1142/S1793524519500918 doi: 10.1142/S1793524519500918
[10]	L. Abia, O. Angulo, J. L$\acute{o}$pez-Marcos, Age-structured population models and their numerical solution, Ecol. Model., 188 (2005), 112–136. https://doi.org/10.1016/j.ecolmodel.2005.05.007 doi: 10.1016/j.ecolmodel.2005.05.007
[11]	A. Din, Y. Li, Ergodic stationary distribution of age-structured HBV epidemic model with standard incidence rate, Nonlinear Dyn., 112 (2024), 9657–9671. https://doi.org/10.1007/s11071-024-09537-4 doi: 10.1007/s11071-024-09537-4
[12]	G. Di Blasio, M. Iannelli, E. Sinestrari, Approach to equilibrium in age structured populations with an increasing recruitment process, J. Math. Biol., 13 (1982), 371–382. https://doi.org/10.1007/BF00276070 doi: 10.1007/BF00276070
[13]	F. Zhang, R. Liu, Y. Chen, Optimal harvesting in a periodic food chain model with size structures in predators, Appl. Math. Optim., 75 (2017), 229–251. https://doi.org/10.1007/s00245-016-9331-y doi: 10.1007/s00245-016-9331-y
[14]	E. Werner, J. Gillian, The ontogenetic niche and species interactions in size-structured populations, Ann. Rev. Ecol. Syst., 15 (1984), 393–425. https://doi.org/10.1146/annurev.es.15.110184.002141 doi: 10.1146/annurev.es.15.110184.002141
[15]	N. Kato, Positive global solutions for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 5 (2000), 191–206. https://doi.org/10.1155/S108533750000035X doi: 10.1155/S108533750000035X
[16]	J. Farkas, T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119–136. https://doi.org/10.1016/j.jmaa.2006.05.032 doi: 10.1016/j.jmaa.2006.05.032
[17]	Y. Lv, Y. Pei, R. Yuan, On a non-linear size-structured population model, Discrete Contin. Dyn. Syst., Ser. B, 25 (2020), 3111–3133. https://doi.org/10.3934/dcdsb.2020053 doi: 10.3934/dcdsb.2020053
[18]	S. Bhattacharya, M. Martcheva, Oscillations in a size-structured prey-predator model, Math. Biosci., 228 (2010), 31–44. https://doi.org/10.1016/j.mbs.2010.08.005 doi: 10.1016/j.mbs.2010.08.005
[19]	M. Mokhtar-Kharroubi, Q. Richard, Spectral theory and time asymptotics of size-structured two-phase population models, Discrete Contin. Dyn. Syst., Ser. B, 25 (2020), 2969–3004. https://doi.org/10.3934/dcdsb.2020048 doi: 10.3934/dcdsb.2020048
[20]	N. Kato, Optimal harvesting for nonlinear size-structured population dynamics, J. Math. Anal. Appl., 324 (2008), 1388–1398. https://doi.org/10.1016/j.jmaa.2008.01.010 doi: 10.1016/j.jmaa.2008.01.010
[21]	Z. He, R. Liu, L. Liu, Optimal harvesting of a size-structured population model in a periodic environment, Acta Math. Appl. Sin., 37 (2014), 145–159.
[22]	Z. He, R. Liu, L. Liu, Optimal harvest rate for a population system modeling periodic environment and body size, Acta Math. Sci. Ser. A Chin. Ed., 34 (2014), 684–690.
[23]	Z. R. He, Y. Liu, An optimal birth control problem for a dynamical population model with size-structure, Nonlinear Anal. Real World Appl., 13 (2012), 1369–1378. https://doi.org/10.1016/j.nonrwa.2011.11.001 doi: 10.1016/j.nonrwa.2011.11.001
[24]	R. Liu, F. Zhang, Y. Chen, Optimal contraception control problems in a nonlinear size-structured vermin model, J. Optim. Theo. Appl., 199 (2023), 1188–1221. https://doi.org/10.1007/s10957-023-02246-9 doi: 10.1007/s10957-023-02246-9
[25]	R. Liu, G. Liu, Optimal harvesting in a unidirectional consumer-resource mutualisms system with size structure in the consumer, Nonlinear Anal.: Model. Control, 27 (2022), 385–411. https://doi.org/10.15388/namc.2022.27.26314 doi: 10.15388/namc.2022.27.26314
[26]	L. M. Abia, O. Angulo, J. C. L$\acute{o}$pez-Marcos, Size-structured population dynamics models and their numerical solutions, Discrete Contin. Dyn. Syst., Ser. B, 4 (2004), 1203–1222. https://doi.org/10.3934/dcdsb.2004.4.1203 doi: 10.3934/dcdsb.2004.4.1203
[27]	R. Liu, G. Liu, Maximum principle for a nonlinear size-structured model of fish and fry management, Nonlinear Anal.: Model. Control, 23 (2018), 533–552. https://doi.org/10.15388/NA.2018.4.5 doi: 10.15388/NA.2018.4.5
[28]	R. Liu, G. Liu, Theory of optimal harvesting for a size structured model of fish, Math. Model. Nat. Phenom., 15 (2020), 1. https://doi.org/10.1051/mmnp/2019006 doi: 10.1051/mmnp/2019006
[29]	V. Barbu, Mathematical methods in optimization of differential systems, Springer Dordrecht, 1994. https://doi.org/10.1007/978-94-011-0760-0
[30]	E. Addai, L. Zhang, J. K. K. Asamoah, J. F. Essel, A fractional order age-specific smoke epidemic model, Appl. Math. Model., 119 (2023), 99–118. https://doi.org/10.1016/j.apm.2023.02.019 doi: 10.1016/j.apm.2023.02.019

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