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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
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show all references

References:
[1]

Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar

[2]

Nat. Resour. Model., 24 (2011), 477-513. doi: 10.1111/j.1939-7445.2011.00099.x.  Google Scholar

[3]

Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-011-0760-0.  Google Scholar

[4]

Math. Biosci., 228 (2010), 31-44. doi: 10.1016/j.mbs.2010.08.005.  Google Scholar

[5]

Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-74001-5.  Google Scholar

[6]

Appl. Math. Lett., 25 (2012), 1041-1044. doi: 10.1016/j.aml.2012.02.067.  Google Scholar

[7]

Math. Biosci., 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2.  Google Scholar

[8]

J. Math. Anal. Appl., 296 (2004), 286-301. doi: 10.1016/j.jmaa.2004.04.052.  Google Scholar

[9]

J. Math. Anal. Appl., 305 (2005), 11-28. doi: 10.1016/j.jmaa.2004.10.002.  Google Scholar

[10]

J. Math. Anal. Appl., 337 (2008), 21-35. doi: 10.1016/j.jmaa.2007.03.082.  Google Scholar

[11]

Nonlinear Anal. Real World Appl., 13 (2012), 1369-1378. doi: 10.1016/j.nonrwa.2011.11.001.  Google Scholar

[12]

Int. J. Biomath., 7 (2014), 1450046, 18 pp. doi: 10.1142/S1793524514500466.  Google Scholar

[13]

Acta Math. Sci. Ser. A Chin. Ed., 34 (2014), 684-690.  Google Scholar

[14]

Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 589-594. doi: 10.3934/dcdsb.2004.4.589.  Google Scholar

[15]

Appl. Math. Lett., 21 (2008), 1090-1094. doi: 10.1016/j.aml.2007.12.006.  Google Scholar

[16]

Agric. Ecosyst. Environ., 115 (2006), 281-284. doi: 10.1016/j.agee.2006.01.001.  Google Scholar

[17]

Abstr. Appl. Anal., 5 (2000), 191-206. doi: 10.1155/S108533750000035X.  Google Scholar

[18]

Int. J. Ecol. Dev., 5 (2006), 6-19. Google Scholar

[19]

J. Math. Anal. Appl., 342 (2008), 1388-1398. doi: 10.1016/j.jmaa.2008.01.010.  Google Scholar

[20]

Abstr. Appl. Anal., 2 (1997), 207-226. doi: 10.1155/S1085337597000353.  Google Scholar

[21]

Abstr. Appl. Anal., 2012 (2012), Art. ID 589202, 14 pp.  Google Scholar

[22]

J. Math. Anal. Appl., 360 (2009), 665-675. doi: 10.1016/j.jmaa.2009.07.005.  Google Scholar

[23]

Acta Math. Sci. Ser. A Chin. Ed., 32 (2012), 90-102.  Google Scholar

[24]

Appl. Math. Comput., 227 (2014), 437-448. doi: 10.1016/j.amc.2013.11.064.  Google Scholar

[25]

J. Math. Anal. Appl., 287 (2003), 557-576. doi: 10.1016/S0022-247X(03)00569-9.  Google Scholar

[26]

Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar

[27]

Science, 179 (1973), 1201-1204. doi: 10.1126/science.179.4079.1201.  Google Scholar

[28]

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]

Princeton University Press, Princeton, New Jersey, 2003.  Google Scholar

[30]

Huazhong University of Science and Technology Press, HuBei (China), 2009. Google Scholar

[31]

Abst. App. Anal., 2014 (2014), Art. ID 396420, 9 pp. doi: 10.1155/2014/396420.  Google Scholar

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