American Institute of Mathematical Sciences

• Previous Article
Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux
• DCDS-B Home
• This Issue
• Next Article
Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype
December  2016, 21(10): 3603-3618. doi: 10.3934/dcdsb.2016112

Optimal contraception control for a nonlinear population model with size structure and a separable mortality

 1 Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, China, China 2 Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000

Received  December 2015 Revised  September 2016 Published  November 2016

This paper is concerned with the problem of optimal contraception control for a nonlinear population model with size structure. First, the existence of separable solutions is established, which is crucial in obtaining the optimal control strategy. Moreover, it is shown that the population density depends continuously on control parameters. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Finally, the conditions of the optimal strategy are derived by means of normal cones and adjoint systems.
Citation: Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112
References:
 [1] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar [2] M. E. Araneda, J. M. Hernández and E. Gasca-Leyva, Optimal harvesting time of farmed aquatic populations with nonlinear size-heterogeneous growth, Nat. Resour. Model., 24 (2011), 477-513. doi: 10.1111/j.1939-7445.2011.00099.x.  Google Scholar [3] V. Barbu, Mathematical Methods in Optimization of Differential Systems (translated and revised from the 1989 Romanian original), Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-011-0760-0.  Google Scholar [4] S. Bhattacharya and M. Martcheva, Oscillation in a size-structured prey-predator model, Math. Biosci., 228 (2010), 31-44. doi: 10.1016/j.mbs.2010.08.005.  Google Scholar [5] B. Ebenman and L. Persson (eds), Size-Structured Populations: Ecology and Evolution, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-74001-5.  Google Scholar [6] M. El-Doma, A size-structured population dynamics model of Daphnia, Appl. Math. Lett., 25 (2012), 1041-1044. doi: 10.1016/j.aml.2012.02.067.  Google Scholar [7] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci., 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2.  Google Scholar [8] Z.-R. He, Optimal birth control of age-dependent competitive species, J. Math. Anal. Appl., 296 (2004), 286-301. doi: 10.1016/j.jmaa.2004.04.052.  Google Scholar [9] Z.-R. He, Optimal birth control of age-dependent competitive species. II. Free horizon problems, J. Math. Anal. Appl., 305 (2005), 11-28. doi: 10.1016/j.jmaa.2004.10.002.  Google Scholar [10] Z.-R. He, J.-S. Cheng and C.-G. zhang, Optimal birth control of age-dependent competitive species. III. Overtaking problem, J. Math. Anal. Appl., 337 (2008), 21-35. doi: 10.1016/j.jmaa.2007.03.082.  Google Scholar [11] Z.-R. He and Y. Liu, An optimal birth control problem for a dynamical population model with size-structure, Nonlinear Anal. Real World Appl., 13 (2012), 1369-1378. doi: 10.1016/j.nonrwa.2011.11.001.  Google Scholar [12] Z.-R. He and R. Liu, Theory of optimal harvesting for a nonlinear size-structured population in periodic environments, Int. J. Biomath., 7 (2014), 1450046, 18 pp. doi: 10.1142/S1793524514500466.  Google Scholar [13] Z. R. He, R. Liu and L. L. Liu, Optimal harvest rate for a population system modeling periodic environment and body size (Chinese), Acta Math. Sci. Ser. A Chin. Ed., 34 (2014), 684-690.  Google Scholar [14] Z.-R. He, M.-S. Wang and Z.-E. Ma, Optimal birth control problem for nonlinear age-structured population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 589-594. doi: 10.3934/dcdsb.2004.4.589.  Google Scholar [15] N. Hritonenko, Y. Yatsenko, R.-U. Goetz and A. Xabadia, Maximum principle for a size-structured model of forest and carbon sequestration management, Appl. Math. Lett., 21 (2008), 1090-1094. doi: 10.1016/j.aml.2007.12.006.  Google Scholar [16] J. Jacob, Rahmini and Sudarmaji, The impact of imposed female sterility on field populations of ricefield rats (Rattus argentiventer), Agric. Ecosyst. Environ., 115 (2006), 281-284. doi: 10.1016/j.agee.2006.01.001.  Google Scholar [17] N. Kato, Positive global solutions for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 5 (2000), 191-206. doi: 10.1155/S108533750000035X.  Google Scholar [18] N. Kato, Linear size-structured population models and optimal harvesting problems, Int. J. Ecol. Dev., 5 (2006), 6-19. Google Scholar [19] N. Kato, Optimal harvesting for nonlinear size-structured population dynamics, J. Math. Anal. Appl., 342 (2008), 1388-1398. doi: 10.1016/j.jmaa.2008.01.010.  Google Scholar [20] N. Kato and H. Torikata, Local existence for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 2 (1997), 207-226. doi: 10.1155/S1085337597000353.  Google Scholar [21] Q. Li, F. Zhang, X. Feng, W. Wang and K. Wang, The permanence and extinction of the single species with contraception control and feedback controls, Abstr. Appl. Anal., 2012 (2012), Art. ID 589202, 14 pp.  Google Scholar [22] Y. Liu and Z.-R. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675. doi: 10.1016/j.jmaa.2009.07.005.  Google Scholar [23] Y. Liu and Z. R. He, Optimal harvesting of a size-structured predator-prey model, Acta Math. Sci. Ser. A Chin. Ed., 32 (2012), 90-102.  Google Scholar [24] Y. Liu and Z.-R. He, Behavioral analysis of a nonlinear three-staged population model with age-size-structure, Appl. Math. Comput., 227 (2014), 437-448. doi: 10.1016/j.amc.2013.11.064.  Google Scholar [25] Z. Luo, Z.-R. He and W.-T. Li, Optimal birth control for age-dependent $n$-dimensional food chain model, J. Math. Anal. Appl., 287 (2003), 557-576. doi: 10.1016/S0022-247X(03)00569-9.  Google Scholar [26] P. Magal and S. Ruan (eds.), Structured-Population Models in Biology and Epidemiology, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [27] T. McMahon, Size and shape in biology: Elastic criteria impose limits on biological proportions, and consequently on metabolic rates, Science, 179 (1973), 1201-1204. doi: 10.1126/science.179.4079.1201.  Google Scholar [28] K. R. Perry, W. M. Arjo, K. S. Bynum and L. A. Miller, GnRH single-injection immunocontraception of black-tailed deer, in Proc. $22^{nd}$ Vertebr. Pest Conf., (eds. R.M. Timm and J.M. O'Brien), Published at Univ. of Calif., Davis (2006). Google Scholar [29] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, New Jersey, 2003.  Google Scholar [30] M. Xia, Q. Liu and S. Li, Functional Analysis and Modern Analysis Tutorial (in Chinese), Huazhong University of Science and Technology Press, HuBei (China), 2009. Google Scholar [31] Q.-J. Xie, Z.-R. He and C.-G. Zhang, Harvesting renewable resources of population with size structure and diffusion, Abst. App. Anal., 2014 (2014), Art. ID 396420, 9 pp. doi: 10.1155/2014/396420.  Google Scholar

show all references

References:
 [1] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar [2] M. E. Araneda, J. M. Hernández and E. Gasca-Leyva, Optimal harvesting time of farmed aquatic populations with nonlinear size-heterogeneous growth, Nat. Resour. Model., 24 (2011), 477-513. doi: 10.1111/j.1939-7445.2011.00099.x.  Google Scholar [3] V. Barbu, Mathematical Methods in Optimization of Differential Systems (translated and revised from the 1989 Romanian original), Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-011-0760-0.  Google Scholar [4] S. Bhattacharya and M. Martcheva, Oscillation in a size-structured prey-predator model, Math. Biosci., 228 (2010), 31-44. doi: 10.1016/j.mbs.2010.08.005.  Google Scholar [5] B. Ebenman and L. Persson (eds), Size-Structured Populations: Ecology and Evolution, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-74001-5.  Google Scholar [6] M. El-Doma, A size-structured population dynamics model of Daphnia, Appl. Math. Lett., 25 (2012), 1041-1044. doi: 10.1016/j.aml.2012.02.067.  Google Scholar [7] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci., 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2.  Google Scholar [8] Z.-R. He, Optimal birth control of age-dependent competitive species, J. Math. Anal. Appl., 296 (2004), 286-301. doi: 10.1016/j.jmaa.2004.04.052.  Google Scholar [9] Z.-R. He, Optimal birth control of age-dependent competitive species. II. Free horizon problems, J. Math. Anal. Appl., 305 (2005), 11-28. doi: 10.1016/j.jmaa.2004.10.002.  Google Scholar [10] Z.-R. He, J.-S. Cheng and C.-G. zhang, Optimal birth control of age-dependent competitive species. III. Overtaking problem, J. Math. Anal. Appl., 337 (2008), 21-35. doi: 10.1016/j.jmaa.2007.03.082.  Google Scholar [11] Z.-R. He and Y. Liu, An optimal birth control problem for a dynamical population model with size-structure, Nonlinear Anal. Real World Appl., 13 (2012), 1369-1378. doi: 10.1016/j.nonrwa.2011.11.001.  Google Scholar [12] Z.-R. He and R. Liu, Theory of optimal harvesting for a nonlinear size-structured population in periodic environments, Int. J. Biomath., 7 (2014), 1450046, 18 pp. doi: 10.1142/S1793524514500466.  Google Scholar [13] Z. R. He, R. Liu and L. L. Liu, Optimal harvest rate for a population system modeling periodic environment and body size (Chinese), Acta Math. Sci. Ser. A Chin. Ed., 34 (2014), 684-690.  Google Scholar [14] Z.-R. He, M.-S. Wang and Z.-E. Ma, Optimal birth control problem for nonlinear age-structured population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 589-594. doi: 10.3934/dcdsb.2004.4.589.  Google Scholar [15] N. Hritonenko, Y. Yatsenko, R.-U. Goetz and A. Xabadia, Maximum principle for a size-structured model of forest and carbon sequestration management, Appl. Math. Lett., 21 (2008), 1090-1094. doi: 10.1016/j.aml.2007.12.006.  Google Scholar [16] J. Jacob, Rahmini and Sudarmaji, The impact of imposed female sterility on field populations of ricefield rats (Rattus argentiventer), Agric. Ecosyst. Environ., 115 (2006), 281-284. doi: 10.1016/j.agee.2006.01.001.  Google Scholar [17] N. Kato, Positive global solutions for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 5 (2000), 191-206. doi: 10.1155/S108533750000035X.  Google Scholar [18] N. Kato, Linear size-structured population models and optimal harvesting problems, Int. J. Ecol. Dev., 5 (2006), 6-19. Google Scholar [19] N. Kato, Optimal harvesting for nonlinear size-structured population dynamics, J. Math. Anal. Appl., 342 (2008), 1388-1398. doi: 10.1016/j.jmaa.2008.01.010.  Google Scholar [20] N. Kato and H. Torikata, Local existence for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 2 (1997), 207-226. doi: 10.1155/S1085337597000353.  Google Scholar [21] Q. Li, F. Zhang, X. Feng, W. Wang and K. Wang, The permanence and extinction of the single species with contraception control and feedback controls, Abstr. Appl. Anal., 2012 (2012), Art. ID 589202, 14 pp.  Google Scholar [22] Y. Liu and Z.-R. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675. doi: 10.1016/j.jmaa.2009.07.005.  Google Scholar [23] Y. Liu and Z. R. He, Optimal harvesting of a size-structured predator-prey model, Acta Math. Sci. Ser. A Chin. Ed., 32 (2012), 90-102.  Google Scholar [24] Y. Liu and Z.-R. He, Behavioral analysis of a nonlinear three-staged population model with age-size-structure, Appl. Math. Comput., 227 (2014), 437-448. doi: 10.1016/j.amc.2013.11.064.  Google Scholar [25] Z. Luo, Z.-R. He and W.-T. Li, Optimal birth control for age-dependent $n$-dimensional food chain model, J. Math. Anal. Appl., 287 (2003), 557-576. doi: 10.1016/S0022-247X(03)00569-9.  Google Scholar [26] P. Magal and S. Ruan (eds.), Structured-Population Models in Biology and Epidemiology, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [27] T. McMahon, Size and shape in biology: Elastic criteria impose limits on biological proportions, and consequently on metabolic rates, Science, 179 (1973), 1201-1204. doi: 10.1126/science.179.4079.1201.  Google Scholar [28] K. R. Perry, W. M. Arjo, K. S. Bynum and L. A. Miller, GnRH single-injection immunocontraception of black-tailed deer, in Proc. $22^{nd}$ Vertebr. Pest Conf., (eds. R.M. Timm and J.M. O'Brien), Published at Univ. of Calif., Davis (2006). Google Scholar [29] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, New Jersey, 2003.  Google Scholar [30] M. Xia, Q. Liu and S. Li, Functional Analysis and Modern Analysis Tutorial (in Chinese), Huazhong University of Science and Technology Press, HuBei (China), 2009. Google Scholar [31] Q.-J. Xie, Z.-R. He and C.-G. Zhang, Harvesting renewable resources of population with size structure and diffusion, Abst. App. Anal., 2014 (2014), Art. ID 396420, 9 pp. doi: 10.1155/2014/396420.  Google Scholar
 [1] Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050 [2] Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283 [3] Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051 [4] Zhipeng Qiu, Huaiping Zhu. Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2703-2728. doi: 10.3934/dcdsb.2016069 [5] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549 [6] Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1967-1990. doi: 10.3934/dcdsb.2020041 [7] Zhigang Ren, Shan Guo, Zhipeng Li, Zongze Wu. Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1579-1594. doi: 10.3934/jimo.2018022 [8] Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289 [9] Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109 [10] Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547 [11] Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507 [12] Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022 [13] Sandra Ricardo, Witold Respondek. When is a control system mechanical?. Journal of Geometric Mechanics, 2010, 2 (3) : 265-302. doi: 10.3934/jgm.2010.2.265 [14] 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 [15] Evgeny I. Veremey, Vladimir V. Eremeev. SISO H-Optimal synthesis with initially specified structure of control law. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 121-138. doi: 10.3934/naco.2017009 [16] Jiawei Dou, Lan-sun Chen, Kaitai Li. A monotone-iterative method for finding periodic solutions of an impulsive competition system on tumor-normal cell interaction. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 555-562. doi: 10.3934/dcdsb.2004.4.555 [17] Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443 [18] Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136 [19] Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173 [20] Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

2020 Impact Factor: 1.327