# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021052
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## Null controllability of a nonlinear age, space and two-sex structured population dynamics model

 1 Université de Fada N'Gourma, Laboratoire LAMI, UJKZ Burkina Faso, DeustoTech Fundación Deusto Avda Universidades, 24, 48007, Bilbao, Basque Country, Spain 2 Département de Mathématiques de la Décision, Laboratoire LAMI, UJKZ Burkina Faso, Université Thomas SANKARA, Burkina Faso

* Corresponding author: Yacouba Simporé

Received  September 2020 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 694126-Dycon)

In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems.

The first problem is related to the total extinction, which means that, we estimate a time $T$ to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time $T,$ and if $A$ is the life span of the individuals, at time $T+A$ one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.

Citation: Yacouba Simporé, Oumar Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021052
##### References:
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##### References:
 [1] B. Ainseba and S. Aniţa, Internal stabilizability for a reaction-diffusion problem modelling a predator-prey system, Nonlinear Anal., 61 (2005), 491-501.  doi: 10.1016/j.na.2004.09.055. [2] B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (2001), 357-368.  doi: 10.1155/S108533750100063X. [3] B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential Integral Equations, 16 (2003), 1369-1384. [4] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [5] I. Boutaayamou and G. Fragnelli, A degenerate population system: Carleman estimates and controllability, Nonlinear Anal., 195 (2020), 29pp. doi: 10.1016/j.na.2019.111742. [6] W. L. Chan and B. Z. Guo., On the semigroups for age-size dependent population dynamics with spatial diffusion, Manuscripia Math., 66 (1989), 161-181.  doi: 10.1007/BF02568489. [7] Y. Echarroudi and L. Maniar, Null controllability of a model in population dynamics, Electron. J. Differential Equations, 2014 (2014), 20pp. [8] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [9] Y. He and B. Ainseba, Exact null controllability of the Lobesia botrana model with diffusion, J. Math. Anal. Appl., 409 (2014), 530-543.  doi: 10.1016/j.jmaa.2013.07.020. [10] N. Hegoburu and S. Aniţa, Null controllability via comparison results for nonlinear age-structured population dynamics, Math. Control Signals Systems, 31 (2019), 38pp. doi: 10.1007/s00498-019-0232-x. [11] N. Hegoburu and M. Tucsnak, Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720.  doi: 10.3934/mcrf.2018030. [12] M. Iannelli and J. Ripol, Two-sex age structured dynamics in a fixed sex-ratio population, Nonlinear Anal. Real World Appl., 13 (2012), 2562-2577.  doi: 10.1016/j.nonrwa.2012.03.002. [13] D. Maity, M. Tucsnak and E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl. (9), 129 (2019), 153-179.  doi: 10.1016/j.matpur.2018.12.006. [14] D. Maity, M. Tucsnak and E. Zuazua, Controllability of a class of infinite dimensional systems with age structure, Control Cybernet., 48 (2019), 231-260. [15] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. [16] Y. Simporé; Controllability of a family of nonlinear population dynamics models, Int. J. Math. Math. Sci., 2021 (2021), 17pp. doi: 10.1155/2021/3581431. [17] Y. Simporé, Null controllability of a nonlinear population dynamics with age structuring and spatial diffusion, in Nonlinear Analysis, Geometry and Applications, Trends Math., Birkhäuser/Springer, Cham, 2020, 1–33. doi: 10.1007/978-3-030-57336-2_1. [18] O. Traore, Null controllability of a nonlinear population dynamics problem, Int. J. Math. Math. Sci., 2006 (2006), 20pp. doi: 10.1155/IJMMS/2006/49279. [19] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985. [20] World Health Organization, World Malaria Report 2018, ISBN: 9789241565653. [21] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Springer-Verlag, New York, 1986. [22] C. Zhao, M. Wang and P. Zhao, Optimal control of harvesting for age-dependent predator-prey system, Math. Comput. Modelling, 42 (2005), 573-584.  doi: 10.1016/j.mcm.2004.07.019.
Illustration of the estimation of $n(x,0,t)$
Illustration of observability inequality in the case where $\max\{a_1, b_1\}+\max\{A-a_2;A-b_2\} = a_1+A-a_2$: The backward characteristics starting from $(a, 0)$ with $a\in (a_2, A)$ (green lines) hits the line $(a = A)$, gets renewed by the renewal condition $(1-\gamma)\beta(a, p)n(x, 0, t)+\gamma\beta(a, p)l(x, 0, t)$ and then enters the observation domain $(a_1, a_2)\cap (b_1, b_2)$. More precisely, the backward characteristics need at most $A-a_2$ time to hits the line $a = A$, get renewed by the renewal condition $(1-\gamma)\beta(a, p)n(x, 0, t)+\gamma\beta(a, p)l(x, 0, t)$ and takes maximum $a_1$ time to enter the observation domain. Thus, at least $T = A-a_2+a_1$ time is needed to obtain the observability inequality. So with the conditions $T > A-a_2+a_1$ and $a_1 < \eta < T,$ all the characteristics starting at $(a, 0)$ with $a\in (a_2, A)$ get renewed by the renewal condition $(1-\gamma)\beta(a, p)n(x, 0, t)+\gamma\beta(a, p)l(x, 0, t)$ in $t\in (0, T-\eta)$ and enter the observation domain
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