适应动力学种群模型中的时间涨落
Time fluctuations in a population model of adaptive dynamics
摘要：
我们研究的动态表型结构的人口在环境中的波动。特别是，使用新的参数从理论的约束和均匀化的Hamilton-Jacobi方程，我们得到的结果与时间振荡的环境中的人口的演变，浓度的发展形式的狄拉克质量，位置的优势性状和它们的演变时间。这样的问题已经在时间均匀环境中研究过了。更确切地说，我们考虑的动态表型结构的人口在不断变化的环境下突变和竞争的单一资源。该数学模型是一个非局部的抛物型方程，具有周期性的时间反应项。在小扩散和快反应的极限下，研究了解的渐近性态。在对反应项的严格假设下，我们证明了该解收敛于一个Dirac质量，该质量随时间的演化是由一个带约束的Hamilton-Jacobi方程和一个作为均匀化极限导出的有效增长/死亡率驱动的.我们还证明了，经过很长一段时间，人口集中在一个性状的有效增长率达到最大值。最后，我们提供了一个例子，表明时间振荡可能会导致严格的渐近总体规模的增加。
英文摘要：
We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton-Jacobi equations with constraints and homogenization, we obtain results about the evolution of populations in environments with time oscillations, the development of concentrations in the form of Dirac masses, the location of the dominant traits and their evolution in time. Such questions have already been studied in time homogeneous environments. More precisely we consider the dynamics of a phenotypically structured population in a changing environment under mutations and competition for a single resource. The mathematical model is a non-local parabolic equation with a periodic in time reaction term. We study the asymptotic behavior of the solutions in the limit of small diffusion and fast reaction. Under concavity assumptions on the reaction term, we prove that the solution converges to a Dirac mass whose evolution in time is driven by a Hamilton-Jacobi equation with constraint and an effective growth/death rate which is derived as a homogenization limit. We also prove that, after long-time, the population concentrates on a trait where the maximum of an effective growth rate is attained. Finally we provide an example showing that the time oscillations may lead to a strict increase of the asymptotic population size.
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