从结构化种群模型到 Kirkpatrick-Barton 模型的宏观极限
Macroscopic limit from a structured population model to the Kirkpatrick-Barton model
AI总结：
研究结合空间抛物算子与表型特征动力学算子的生态学模型，证明解的存在唯一性，并利用 Wasserstein 距离收缩与抛物估计，证明其向 Kirkpatrick-Barton 模型解的宏观收敛性。
摘要：
我们考虑一个生态学模型，其中种群由空间变量和表型特征变量进行结构化。该模型将空间变量上的抛物算子与特征变量上的动力学算子相结合。我们证明了该模型解的存在性，并表明这些解是唯一的。模型中代表有性繁殖效应的动力学算子满足 Tanaka 型不等式：它意味着表型特征空间中 Wasserstein 距离的收缩。我们将这一收缩论证与控制解的空间正则性的抛物估计相结合，证明了种群规模和平均表型特征向 Kirkpatrick-Barton 模型（演化生态学中一个成熟的模型）解的收敛性。具体而言，在高繁殖率下，我们为动力学模型解的矩提供了显式收敛估计。
英文摘要：
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.
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