Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrodifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.

We study an equation structured by age and a phenotypic trait describing the growth process of a population subject to aging, competition between individuals, and mutations. This leads to a renewal equation which occurs in many evolutionary biology problems. We aim to describe precisely the asymptotic behavior of the solution, to infer properties that illustrate the concentration and adaptive dynamics of such a population. This work is a continuation of [38] where the case without mutations is considered. When mutations are taken into account, it is necessary to control the corrector which is the main novelty of the present paper.Our approach consists in defining, by the Hopf transform, a Hamilton-Jacobi equation with an effective Hamiltonian as in homogenization problems. Its solution carries the singular part of the limiting density (typically Dirac masses) and the corrector defines the weights. The main new result of this paper is to prove that the corrector is uniformly bounded, using only the global Lipschitz and semi-convexity estimates for the viscosity solution of the Hamilton-Jacobi equation. We also establish the limiting equation satisfied by the corrector. To the best of our knowledge, this is the first example where such bounds can be proved in such a context.

We call pattern any non-constant solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [20] and Matano [50] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the stability of patterns, and the C 1 norm of the nonlinearity.In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L 2 -norm of the gradient of a solution is controlled by the L 2 -norm of f (u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates.Our approach relies on the introduction of what we call the Robincurvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions.Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [54] and to extend some results on De Giorgi's conjecture to a larger class of domains.

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