We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solution to the unique positive and periodic solution of the problem, we study the influence of different factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the influence of the diffusion rate, the period length and the time variance of the environment fluctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the influence of a periodic treatment. In this framework, we show that the population might benefit from a intermediate rate of mutation.

中文翻译：

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时间周期环境中表型结构种群突变的影响

我们研究一个抛物线型Lotka-Volterra方程，其中一个积分项代表竞争和时间周期增长率。该模型表示时间周期环境中的特征结构化种群。在显示出对问题的唯一正解和周期解的收敛性之后，我们研究了不同因素对平均极限总体的影响。由于此数量与某个特征值相反，因此我们能够研究扩散速率，周期长度和环境波动的时间变化的影响。如果模型代表在定期治疗的影响下癌细胞的数量，我们还将在癌症框架内对结果进行生物学解释。在这个框架中