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

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.

References in corpus (4)

• On a model of a population with variable motility
• Tanaka Theorem for Inelastic Maxwell Models
• Existence and qualitative properties of travelling waves for an epidemiological model with mutations
• Population structured by a space variable and a phenotypical trait

Cited by in corpus (7)

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• Ergodicity of the Fisher infinitesimal model with quadratic selection
• Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection
• Dynamics of dirac concentrations in the evolution of quantitative alleles with sexual reproduction
• Evolutionary dynamics of complex traits in sexual populations in a strongly heterogeneous environment: how normal?
• On the Fisher infinitesimal model without variability
• The emergence of a birth-dependent mutation rate: causes and consequences