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A Hamilton-Jacobi approach for the evolutionary dynamics of a model with gene transfer: characterizing monomorphic dynamics for non-concave fitness functions

Alejandro Gárriz, Sepideh Mirrahimi

Abstract

We study the asymptotic behavior of an integro-dierential equation describing the evolutionary adaptation of a population structured by a phenotypic trait. The model takes into account mutation, selection, horizontal gene transfer and competition. Previous works, based on the numerical studies or theoretical study of the corresponding stationary problem, have shown that the dynamics of the solutions are rich and we may expect several qualitative outcomes. In this article, we characterize the dynamics of the solution in two regimes: 1) a situation where the solution concentrates around a dominant trait, evolving gradually to a trait determined by a balance between selection and horizontal gene transfer; 2) a situation where the solution concentrates around a dominant trait which evolves gradually to a maladapted trait such that the population becomes extinct (a situation known as the evolutionary suicide). Our analysis is based on an approach involving Hamilton-Jacobi equations with constraint. Previously, the solutions to such equations were characterized for globally concave growth rates. Here, we extend this approach to situations where the growth rate is not globally concave.

A Hamilton-Jacobi approach for the evolutionary dynamics of a model with gene transfer: characterizing monomorphic dynamics for non-concave fitness functions

Abstract

We study the asymptotic behavior of an integro-dierential equation describing the evolutionary adaptation of a population structured by a phenotypic trait. The model takes into account mutation, selection, horizontal gene transfer and competition. Previous works, based on the numerical studies or theoretical study of the corresponding stationary problem, have shown that the dynamics of the solutions are rich and we may expect several qualitative outcomes. In this article, we characterize the dynamics of the solution in two regimes: 1) a situation where the solution concentrates around a dominant trait, evolving gradually to a trait determined by a balance between selection and horizontal gene transfer; 2) a situation where the solution concentrates around a dominant trait which evolves gradually to a maladapted trait such that the population becomes extinct (a situation known as the evolutionary suicide). Our analysis is based on an approach involving Hamilton-Jacobi equations with constraint. Previously, the solutions to such equations were characterized for globally concave growth rates. Here, we extend this approach to situations where the growth rate is not globally concave.

Paper Structure

This paper contains 22 sections, 23 theorems, 290 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model and question
  3. Biological motivation and the expected outcomes
  4. State of the art
  5. An adimensional parameterization of the problem
  6. Assumptions
  7. Main results
  8. To what extent can the assumptions be relaxed?
  9. Organization of the article
  10. Uniqueness and regularity for the Hamilton-Jacobi equation with constraint under a local concavity assumption
  11. Proof of Theorem \ref{['thm:section_Dyn_Prog_1']}
  12. Proof of Theorem \ref{['thm:section_Dyn_Prog_2']}
  13. Proof of Theorem \ref{['thm:limit_epsilon']}
  14. Regularity estimates
  15. Convergence to the Hamilton-Jacobi equation
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let conditions hyp:R, eq:hypothesis_H, eq:hypothesis_u_0 and eq:hypothesis_rho be satisfied. As $\varepsilon\to 0$ and along subsequences, $n_\varepsilon$ converges in $L^\infty(w \ast(\mathbb{R}^+);\mathcal{M}^1(\mathbb{R}))$ to a measure $n\in L^\infty(\mathbb{R}^+;\mathcal{M}^1(\mathbb{R}))$, $\r The solutions $u_{\varepsilon }$ converge locally uniformly to a continuous function $u$ that is a

Figures (6)

  • Figure 1: A collection of rectangles $\{\mathcal{R}_i\}_{i=1}^7$, in grey, that contain the curve ${\bar{z}}(t)$, in dashed-black. Each horizontal black line is a set $\mathcal{U}(t_i)$, for $i=1,...,7$. In dotted-black, the curve $\omega_1(t)$ and the vertical line corresponds to $\{ z=\mu \}$.
  • Figure 2: Two examples of $F(0,z)$. The second derivative $\partial_{zz}F(0,z)$ is in dashed-black, in order to appreciate concavity sets. As we can see, the value ${\bar{z}}(0)$ is to the right of the dotted vertical line representing $z=\mu$. The values of the parameters are $\tau=0,5$ and $\mu=1,7$. Note that in both cases $\mu<{\bar{z}}(0)<\mu_1$ and there is no remarkable difference between the functions $\partial_{zz}F(0,z)$.
  • Figure 3: Fitness functions $F$ (red) with ${\bar{z}}(t)=\mu<\mu_1$. The point where $F=0$ is where $z=\mu$ (green).
  • Figure 4: Fitness functions $F$ (red) with ${\bar{z}}(t)=\mu_1$. The points where $F=0$ are now $z=\mu_1$ and a second point $z=\mu_1-d_1$ (green), where $d_1$ comes from \ref{['eq:d1']}.
  • Figure 5: Fitness functions $F$ (red) with ${\bar{z}}(t)=\mu>\mu_1$. A set $J$ (blue) appears where $F>0$, far away from the point $z=\mu$ (green).
  • ...and 1 more figures

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3: The globally concave case
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 34 more