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Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

Gaël Raoul

TL;DR

The work analyzes a population density $n(t,x,y)$ structured by space $x$ and trait $y$, governed by a diffusion in $x$ and a reproducing, trait-diffusing operator in $y$ (the SIM). It first establishes global existence and uniqueness of SIM solutions with uniform tail and regularity controls, then leverages a Tanaka-type Wasserstein contraction to show that for large $\gamma$ the trait distribution concentrates near a Gaussian with mean $Z(t,x)$ and variance $A$, i.e. $n\approx N\,\Gamma_A(\cdot-Z)$. The macroscopic fields $(N,Z)$ satisfy a closed Kirkpatrick-Barton system (KBM) up to explicit corrections $\varphi_{N,\gamma}$ and $\varphi_{Z,\gamma}$, with quantitative bounds $W_2(\tilde n, \Gamma_A(\cdot-Z)) \le C/\gamma^{\theta}$ after an initial layer, and convergence of $(N,Z)$ to the KBM solution as $\gamma\to\infty$. This provides a rigorous link between a mesoscopic, sexually reproducing structured population model and the classical ecological KBM, with implications for understanding climate-driven trait dynamics and range shifts.

Abstract

We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We prove the existence of solutions to that model, and show that these solutions are unique. The kinetic operator present in the model, that represents the effect of sexual reproductions, satisfies a Tanaka-type inequality: it implies a contraction of the Wasserstein distance in the space of phenotypic traits. We combine this contraction argument with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology. Specifically, at high reproductive rates, we provide explicit convergence estimates for the moments of solutions of the kinetic model.

Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

TL;DR

The work analyzes a population density structured by space and trait , governed by a diffusion in and a reproducing, trait-diffusing operator in (the SIM). It first establishes global existence and uniqueness of SIM solutions with uniform tail and regularity controls, then leverages a Tanaka-type Wasserstein contraction to show that for large the trait distribution concentrates near a Gaussian with mean and variance , i.e. . The macroscopic fields satisfy a closed Kirkpatrick-Barton system (KBM) up to explicit corrections and , with quantitative bounds after an initial layer, and convergence of to the KBM solution as . This provides a rigorous link between a mesoscopic, sexually reproducing structured population model and the classical ecological KBM, with implications for understanding climate-driven trait dynamics and range shifts.

Abstract

We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We prove the existence of solutions to that model, and show that these solutions are unique. The kinetic operator present in the model, that represents the effect of sexual reproductions, satisfies a Tanaka-type inequality: it implies a contraction of the Wasserstein distance in the space of phenotypic traits. We combine this contraction argument with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology. Specifically, at high reproductive rates, we provide explicit convergence estimates for the moments of solutions of the kinetic model.

Paper Structure

This paper contains 20 sections, 13 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Main results and organisation of the paper
  3. Existence and uniqueness of solutions of (SIM)
  4. Macroscopic limits of solutions of (SIM)
  5. Biological interpretation of the model and impact for ecology
  6. Preliminary: equations satisfied by solutions of the (SIM)
  7. Existence of solutions for (SIM)
  8. Macroscopic limits of solutions of (SIM)
  9. Tail estimates uniform in $\gamma>0$ for solutions of (SIM)
  10. Regularity estimates uniform in $\gamma>0$ for $N$ and $Z$
  11. Distance of solutions of (SIM) to local Maxwellians
  12. Macroscopic limit from (SIM) to (KBM)
  13. Appendix
  14. Wasserstein distances
  15. Properties of the Infinitesimal operator
  16. ...and 5 more sections

Key Result

Proposition 2.2

Let $y_{opt}\in W^{1,\infty}(\mathbb R_+\times\mathbb T^d)$, $n^0$ satisfying Assumption Assumption3, $A>0$, and $\gamma>2+A+\|y_{opt}\|_{L^\infty(\mathbb R_+\times\mathbb T^d)}$. There is a unique global solution $n$ of (SIM) with the initial data $n^0$, in the sense of definition def:sol. More pre while for any $T>0$, there is $C_{\gamma,T}>0$ such that Moreover, for $(t,x,y)\in \mathbb R_+\tim

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 4.1
  • Remark 4.2
  • proof : Proof of Proposition \ref{['prop:moment4']}
  • ...and 20 more