# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021052
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 & Related Fields, doi: 10.3934/mcrf.2021052
##### References:

show all references

##### References:
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
 [1] Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031 [2] Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic & Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873 [3] Gigi Thomas, Edward M. Lungu. A two-sex model for the influence of heavy alcohol consumption on the spread of HIV/AIDS. Mathematical Biosciences & Engineering, 2010, 7 (4) : 871-904. doi: 10.3934/mbe.2010.7.871 [4] Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021037 [5] Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217 [6] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [7] Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 [8] Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014 [9] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [10] Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065 [11] Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021004 [12] Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775 [13] Wei Feng, Xin Lu, Richard John Donovan Jr.. Population dynamics in a model for territory acquisition. Conference Publications, 2001, 2001 (Special) : 156-165. doi: 10.3934/proc.2001.2001.156 [14] Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623 [15] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [16] Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 [17] Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037 [18] Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521 [19] Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949 [20] Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

2020 Impact Factor: 1.284