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Ergodicity of the Fisher infinitesimal model with quadratic selection

Vincent Calvez, Thomas Lepoutre, David Poyato

TL;DR

This work tackles the case of quadratic selection by a global approach and establishes uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile, which can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

Abstract

We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogues, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

Ergodicity of the Fisher infinitesimal model with quadratic selection

TL;DR

This work tackles the case of quadratic selection by a global approach and establishes uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile, which can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

Abstract

We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogues, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

Paper Structure

This paper contains 23 sections, 25 theorems, 276 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and strategy of the proof of Theorem \ref{['T-main']}
  3. Some properties of the sexual reproduction operator
  4. Incompatibility of Wasserstein metric with multiplicative operators
  5. Log-Lipschitz contraction estimate
  6. Brief description of our strategy
  7. Preliminaries
  8. Perfect binary trees
  9. Gaussian solutions
  10. Some properties of the operator $\mathcal{T}$
  11. Emergence of Gaussian tails
  12. Propagation of quadratic and exponential moments
  13. Reformulating the recursion
  14. A change of variables across the binary tree
  15. Probabilistic reinterpretation
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $\alpha\in \mathbb{R}_+^*$ and set any initial non-negative measure $F_0\in \mathcal{M}_+(\mathbb{R}^d)$. The solution $\{F_n\}_{n\in \mathbb{N}}$ to the time-discrete problem E-sexual-reproduction-time-discrete verifies that the growth rates $\Vert F_n\Vert_{L^1(\mathbb{R}^d)}/\Vert F_{ and the variance $\mathop{\mathrm{\boldsymbol{\sigma}}}\nolimits_\alpha^2\in \mathbb{R}_+^*$ is the

Figures (11)

  • Figure 1: Parameter $\mathop{\mathrm{\boldsymbol{k}}}\nolimits_\alpha$ against parameter $\alpha$.
  • Figure 2: Overall map of the proof of Theorem \ref{['T-main']}
  • Figure 3: Perfect binary tree $\mathop{\mathrm{\mathsf{T}}}\nolimits^3$
  • Figure 4: Plot of eigenvalue $\mathop{\mathrm{\boldsymbol{\lambda}}}\nolimits_\alpha$ and variance at equilibrium $\mathop{\mathrm{\boldsymbol{\sigma}}}\nolimits_\alpha^2$ against $\alpha$ for $d=1$. The case $\alpha = 0$ corresponds to absence of selection, which is conservative ($\mathop{\mathrm{\boldsymbol{\lambda}}}\nolimits_{\alpha=0} = 1$), and the variance at equilibrium $\mathop{\mathrm{\boldsymbol{\sigma}}}\nolimits_{\alpha=0}^2$ is twice the variance of the mixing kernel in $\mathcal{B}$.
  • Figure 5: The ratio $\mathop{\mathrm{\boldsymbol{r}}}\nolimits_\alpha$ against parameter $\alpha$.
  • ...and 6 more figures

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3: About the assumption of quadratic selection
  • Remark 1.4: About the choice of metric
  • Remark 1.5: About the positivity of $\alpha$
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Definition 2.3: Normalization of multiplicative operator
  • Example 2.4
  • ...and 54 more