受选择和突变影响的种群的生存准则;应用于时间分段恒定环境
Survival criterion for a population subject to selection and mutations ; Application to temporally piecewise constant environments
摘要：
我们研究了一个抛物型Lotka-Volterra型方程，它描述了由表型性状构成的种群的演化，在突变和竞争资源的影响下，由非局部反馈建模。小突变的极限由具有约束的Hamilton-Jacobi方程表征，该方程描述了群体对某些性状的集中。这一结果已经在Barles-Perthame 2008、Barles-Mirrahimi-Perthame 2009、Lorz-Mirrahimi-Perthame 2011中在时间齐次环境中建立，当时通过对增长率或初始数据的假设来确保种群的渐近持久性。在这里，我们放松这些假设，将研究扩展到种群可能在极限灭绝的情况。为此目的，我们提供了条件的初始数据的渐近命运的人口。最后，我们展示了如何在时间均匀的环境中进行这项研究，可以考虑时间分段恒定的环境。
英文摘要：
We study a parabolic Lotka-Volterra type equation that describes the evolution of a population structured by a phenotypic trait, under the effects of mutations and competition for resources modelled by a nonlocal feedback. The limit of small mutations is characterized by a Hamilton-Jacobi equation with constraint that describes the concentration of the population on some traits. This result was already established in Barles-Perthame 2008, Barles-Mirrahimi-Perthame 2009, Lorz-Mirrahimi-Perthame 2011 in a time-homogenous environment, when the asymptotic persistence of the population was ensured by assumptions on either the growth rate or the initial data. Here, we relax these assumptions to extend the study to situations where the population may go extinct at the limit. For that purpose, we provide conditions on the initial data for the asymptotic fate of the population. Finally, we show how this study for a time-homogenous environment allows to consider temporally piecewise constant environments.
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