Abstract
This paper investigates the optimal contraception control for a nonlinear size-structured population model with three kinds of mortality rates: intrinsic, intra-competition and female sterilant. First, we transform the model to a system of two subsystems, and establish the existence of a unique non-negative solution by means of frozen coefficients and fixed point theory, and show the continuous dependence of the population density on control variable. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Next, necessary optimality conditions of first order are established in the form of an Euler–Lagrange system by the use of tangent-normal cone technique and adjoint system. Moreover, a numerical result for the optimal control strategy is presented. Our conclusions would be useful for managing the vermin.
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References
Aniţa, S.: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Dordrecht (2000)
Aniţa, L.I., Aniţa, S., Arnăutu, V.: Optimal harvesting for periodic age-dependent population dynamics with logistic term. Appl. Math. Comput. 215, 2701–2715 (2009)
Aniţa, S., Iannelli, M., Kim, M.Y., Park, E.J.: Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 58(5), 1648–1666 (1998)
Araneda, M.E., Hernández, J.M., Gasca-Leyva, E.: Optimal harvesting time of farmed aquatic populations with nonlinear size-heterogeneous growth. Nat. Resour. Model. 24, 477–513 (2011)
Banaś, J., Caballero, J., Rocha, J., Sadarangani, K.: Monotonic solutions of a class of quadratic integral equations of Volterra type. Comput. Math. Appl. 49, 943–952 (2005)
Bhattacharya, S., Martcheva, M.: Oscillation in a size-structured prey–predator model. Math. Biosci. 228, 31–44 (2010)
Ebenman, B., Persson, L.: Size-Structured Populations: Ecology and Evolution. Springer, Berlin (1988)
Eucario, G.L.: Optimal harvesting time in a size-heterogeneous population. Ecol. Model. 210, 161–168 (2008)
Fister, K.R., Lenhart, S.: Optimal harvesting in an age-structured predator–prey model. Appl. Math. Optim. 54, 1–15 (2006)
He, Z.R., Cheng, J.S., Zhang, C.-G.: Optimal birth control of age-dependent competitive species: III. Overtaking problem. J. Math. Anal. Appl. 337, 21–35 (2008)
He, Z.R., Liu, R., Liu, L.L.: Optimal harvest rate for a population system modeling periodic environment and body size. Acta Math. Sci. Ser. A Chin. Ed. 34, 684–690 (2014). Chinese
He, Z.R., Liu, R.: Theory of optimal harvesting for a nonlinear size-structured population in periodic environments. Int. J. Biomath. 4, 201–208 (2014)
He, Z.R., Liu, Y.: An optimal birth control problem for a dynamical population model with size-structure. Nonlinear Anal. Real World Appl. 13, 1369–1378 (2012)
Hritonenko, N., Yatsenko, Yu., Goetz, R., Xabadia, A.: Maximum principle for a size-structured model of forest and carbon sequestration management. Appl. Math. Lett. 21, 1090–1094 (2008)
Hritonenko, N., Yatsenko, Yu., Goetz, R., Xabadia, A.: A bang-bang regime in optimal harvesting of size-structured populations. Nonlinear Anal. 71, e2331–e2336 (2009)
Hughes, T.R., Connell, J.H.: Population dynamics based on size or age? Areef-coral analysis. Am. Nat. 129(6), 818–829 (1987)
Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori, Pisa (1995)
Jacob, J., Rahmini, S.: The impact of imposed female sterility on filed populations of ricefiled rats (Rattus argentiventer). Agric. Ecosyst. Environ. 115, 281–284 (2006)
Kato, N.: Maximum principle for optimal harvesting in linear size-structured population. Math. Popul. Stud. 15, 123–136 (2008)
Kato, N.: Optimal harvesting for nonlinear size-structured population dynamics. J. Math. Anal. Appl. 324, 1388–1398 (2008)
Li, Q., Zhang, F., Feng, X., et al.: The permanence and extinction of the single species with contraception control and feedback controls. Abstr. Appl. Anal. (2012). doi:10.1155/2012/589202
Liu, Y., He, Z.R.: Optimal harvesting of a size-structured predator–prey model. Acta Math. Sci. Ser. A Chin. Ed. 32, 90–102 (2012)
Liu, Y., He, Z.R.: Behavioral analysis of a nonlinear three-staged population model with age-size-structure. Appl. Math. Comput. 227, 437–448 (2014)
Liu, R., Liu, G.R.: Optimal birth control problems for a nonlinear vermin population model with size-structure. J. Math. Anal. Appl. 449, 265–291 (2017)
Liu, R., Zhang, F.Q., Chen, Y.M.: Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discret. Contin. Dyn. Syst. Ser. B 21, 3603–3618 (2016)
Luo, Z.X.: Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model. Appl. Math. Comput. 186, 1742–1752 (2007)
Magal, P., Ruan, S., et al.: Structured-Population Models in Biology and Epidemiology. Springer, Berlin (2008)
Metz, J.A.J, Dickmann, O.: The dynamics of physiologically structured population. In: Lecture Notes in Biomathematics, vol. 68. Springer, Berlin (1986)
Zhang, F.Q., Liu, R., Chen, Y.M.: Optimal harvesting in a periodic food chain model with size structures in predators. Appl. Math. Optim. 75, 229–251 (2017)
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This work was supported partially by the National Natural Science Foundation of China (Nos. 11471197, 11571210, 11501339).
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Liu, R., Liu, G. Optimal Contraception Control for a Nonlinear Vermin Population Model with Size-Structure. Appl Math Optim 79, 231–256 (2019). https://doi.org/10.1007/s00245-017-9428-y
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DOI: https://doi.org/10.1007/s00245-017-9428-y