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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

N. Kato, Linear size-structured population models and optimal harvesting problems, Int. J. Ecol. Dev., 5 (2006), 6-19.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[29]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, New Jersey, 2003.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

N. Kato, Linear size-structured population models and optimal harvesting problems, Int. J. Ecol. Dev., 5 (2006), 6-19.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[29]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, New Jersey, 2003.

[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.

[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.

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