SIAM Journal on Applied Mathematics, Volume 80, Issue 6, Page 2420-2447, January 2020.
We analyze a nonlocal partial differential equation (PDE) model describing the dynamics of adaptation of a phenotypically structured population, under the effects of mutation and selection, in a changing environment. Previous studies have analyzed the large-time behavior of such models, with particular forms of environmental changes---either linearly changing or periodically fluctuating. We use here a completely different mathematical approach, which allows us to consider very general forms of environmental variations and to give an analytic description of the full trajectories of adaptation, including the transient phase, before a stationary behavior is reached. The main idea behind our approach is to study a bivariate distribution of two “fitness components" that contains enough information to describe the distribution of fitness at any time. This distribution solves a degenerate parabolic equation that is dealt with by defining a multidimensional cumulant generating function associated with the distribution and solving the associated transport equation. We apply our results to several examples and check their accuracy using stochastic individual-based simulations as a benchmark. These examples illustrate the importance of being able to describe the transient dynamics of adaptation to understand the development of drug resistance in pathogens.

中文翻译：

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在一般临时变化的环境中适应

SIAM应用数学杂志，第80卷，第6期，第2420-2447页，2020年1月。
我们分析了一个非局部偏微分方程（PDE）模型，该模型描述了在变化的环境中，在突变和选择的影响下，表型结构化种群适应的动力学。先前的研究已经分析了此类模型的长时间行为，以及特定形式的环境变化-线性变化或周期性波动。在这里，我们使用完全不同的数学方法，该方法允许我们考虑环境变化的非常一般的形式，并在达到平稳行为之前对包括过渡阶段在内的整个适应轨迹进行分析描述。我们方法背后的主要思想是研究两个“健身成分”的二元分布 其中包含足够的信息，可以随时描述健身的分布情况。这种分布解决了退化的抛物线方程，通过定义与该分布相关的多维累积量生成函数并求解相关的输运方程来处理该退化的抛物线方程。我们将结果应用于多个示例，并使用基于个体的随机模拟作为基准来检查其准确性。这些例子说明了能够描述适应过程的动态变化以了解病原体耐药性发展的重要性。我们将结果应用于多个示例，并使用基于个体的随机模拟作为基准来检查其准确性。这些例子说明了能够描述适应过程的动态变化以了解病原体耐药性发展的重要性。我们将结果应用于多个示例，并使用基于个体的随机模拟作为基准来检查其准确性。这些例子说明了能够描述适应过程的动态变化以了解病原体耐药性发展的重要性。