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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 & 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, (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. 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, (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. 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, (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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. Google Scholar

[19]

N. Kato, Optimal harvesting for nonlinear size-structured population dynamics,, J. Math. Anal. Appl., 342 (2008), 1388. 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. 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). 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. 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. 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. 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. 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, (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. 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., (2006). Google Scholar

[29]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (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, (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). 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, (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. 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, (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. 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, (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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. Google Scholar

[19]

N. Kato, Optimal harvesting for nonlinear size-structured population dynamics,, J. Math. Anal. Appl., 342 (2008), 1388. 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. 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). 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. 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. 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. 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. 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, (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. 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., (2006). Google Scholar

[29]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (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, (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). doi: 10.1155/2014/396420. Google Scholar

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