Constrained control of gene-flow models
Mazari, Idriss; Ruiz-Balet, Domènec; Zuazua, Enrique (2022), Constrained control of gene-flow models, Annales de l'Institut Henri Poincaré (C) Analyse non linéaire. 10.4171/aihpc/52
TypeArticle accepté pour publication ou publié
Journal nameAnnales de l'Institut Henri Poincaré (C) Analyse non linéaire
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Departamento de Matemáticas [Madrid]
Friedrich-Alexander Universität Erlangen-Nürnberg [FAU]
Abstract (EN)In ecology and population dynamics, gene flow refers to the transfer of a trait (e.g. genetic material) from one population to another. This phenomenon is of great relevance in studying the spread of diseases or the evolution of social features, such as languages. From the mathematical point of view, gene flow is modeled using bistable reaction–diffusion equations. The unknown is the proportion ppp of the population that possesses a certain trait, within an overall population NNN. In such models, gene flow is taken into account by assuming that the population density NNN depends either on ppp (if the trait corresponds to fitter individuals) or on the location xxx (if some zones in the domain can carry more individuals). Recent applications stemming from mosquito-borne-disease control problems or from the study of bilingualism have called for the investigation of the controllability properties of these models. At the mathematical level, this corresponds to boundary control problems and, since we are working with proportions, the control uuu has to satisfy the constraint 0≤u≤10\leq u \leq 10≤u≤1. In this article, we provide a thorough analysis of the influence of the gene-flow effect on boundary controllability properties. We prove that, when the population density NNN only depends on the trait proportion ppp, the geometry of the domain is the only criterion that has to be considered. We then tackle the case of population densities NNN varying in xxx. We first prove that, when NNN varies slowly in xxx and when the domain is narrow enough, controllability always holds. This result is proved using a robust domain perturbation method. We then consider the case of sharp fluctuations in NNN: we first give examples that prove that controllability may fail. Conversely, we give examples of heterogeneities NNN such that controllability will always be guaranteed: in other words the controllability properties of the equation are very strongly influenced by the variations of NNN. All negative controllability results are proved by showing the existence of nontrivial stationary states, which act as barriers. The existence of such solutions and the methods of proof are of independent interest. Our article is completed by several numerical experiments that confirm our analysis.
Subjects / KeywordsControl; reaction-diffusion equations; bistable equations; spatial heterogeneity; geneflow models; staircase method; non-controllability
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