[Paper Review] Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection

Gaël Raoul|arXiv (Cornell University)|Apr 13, 2021

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TL;DR

This paper establishes exponential convergence to steady states in a population genetics model with sexual reproduction and weak selection, using Wasserstein distance inequalities and tail estimates. It proves that solutions converge exponentially fast to locally unique steady states when selection strength α is small, leveraging the contraction properties of the reproduction operator and stability analysis under selection perturbations.

ABSTRACT

We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady-states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady-states.

Motivation & Objective

• To analyze the long-time dynamics of a population structured by phenotypic traits under sexual reproduction and selection.
• To establish exponential convergence of solutions to steady states when selection strength α is small.
• To provide a stability analysis of steady states using Wasserstein distances rather than regularity-based methods.
• To balance the destabilizing effect of selection with the contracting effect of the sexual reproduction operator.
• To prove local uniqueness and exponential convergence to steady states under small α, using uniform lower bounds and tail estimates.

Proposed method

• Uses a non-linear integral operator to model sexual reproduction, with offspring traits drawn from a Gaussian centered at the average of parental traits.
• Combines the reproduction operator with a multiplicative selection term (1 + αa(x)) to model fitness differences.
• Applies Wasserstein distance W₂ to measure the distance between probability measures of trait distributions over time.
• Employs Wasserstein contraction inequalities from the reproduction operator, derived from Raoul (2017), to control solution evolution.
• Uses uniform lower bounds on population densities and tail estimates to control the destabilizing effect of selection.
• Analyzes the dynamics via macroscopic quantities such as centers of mass and selection-weighted moments, estimating their differences using functional inequalities.

Experimental results

Research questions

• RQ1 How does the population distribution evolve over time in a sexually reproducing population with weak selection?
• RQ2 Under what conditions does the system converge exponentially to a steady state?
• RQ3 Can the destabilizing effect of selection be counterbalanced by the contracting effect of the sexual reproduction operator?
• RQ4 What role do Wasserstein distances play in quantifying convergence to steady states in this non-local, non-linear model?
• RQ5 How does the local uniqueness of steady states relate to the strength of selection α?

Key findings

• For small α > 0, solutions to the kinetic equation converge exponentially to a steady state, with convergence rate governed by F′(Z̄)α, where F′(Z̄) < 0.
• The steady state is locally unique, as established by Lemma 2.2 and confirmed through the contraction argument.
• Exponential convergence is proven via a Lyapunov-type argument using the quantity √α X(t) + |Zₙ∘φₙ(t) − Zₘ∘φₘ(t)|, which decays exponentially after time t₀ = −C ln α / α.
• The Wasserstein distance between two solutions satisfies W₂(𝒩(t,·), 𝒪(t,·)) ≤ C exp((F′(Z̄) + 𝒪(√α))αt), showing exponential decay to the steady state.
• Tail estimates and uniform lower bounds on distributions are essential to control the selection term’s destabilizing influence.
• The method avoids reliance on regularity estimates (e.g., derivatives of U = −σ log n), instead using transport-based metrics, offering a new analytical framework for such models.