Mazari I, Ruiz-Balet D, Zuazua Iriondo E (2024)

**Publication Language:** English

**Publication Status:** Accepted

**Publication Type:** Unpublished / Preprint

**Future Publication Type:** Journal article

**Publication year:** 2024

**Publisher:** Ann. Inst. Henri Poincare (C) Anal. Non Lineaire

**URI:** https://dcn.nat.fau.eu/wp-content/uploads/ControlHeterogeneousV5.pdf

**DOI:** 10.4171/aihpc/52

**Open Access Link:** https://dcn.nat.fau.eu/wp-content/uploads/ControlHeterogeneousV5.pdf

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 modelled using
bistable reaction-diffusion equations. The unknown is the proportion p
of the population that possesses a certain trait, within an overall
population N. In such models, gene-flow is taken into account by
assuming that the population density N depends either on p (if the trait
corresponds to fitter individuals) or on the location x (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 u has to satisfy the constraints 0≤u≤1 0 \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 N only depends on the trait proportion p,
the geometry of the domain is the only criterion that has to be
considered. We then tackle the case of population densities N varying in
x. We first prove that, when N varies slowly in x 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 N: we first give examples that prove that
controllability may fail. Conversely, we give examples of
heterogeneities N such that controllability will always be guaranteed:
in other words the controllability properties of the equation are very
strongly in uenced by the variations of N. All negative controllability
results are proved by showing the existence of non-trivial 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.

Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

**APA:**

Mazari, I., Ruiz-Balet, D., & Zuazua Iriondo, E. (2024). *Constrained control of gene-flow models. *(Unpublished, Accepted).

**MLA:**

Mazari, Idriss, Domènec Ruiz-Balet, and Enrique Zuazua Iriondo. *Constrained control of gene-flow models.* Unpublished, Accepted. 2024.

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