发表机构
Laboratoire de Mathématiques Jean Leray UMR 6629 Université de Nantes（南特大学让·勒雷数学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对随机环境下选择梯度计算的数学难题，建立了带随机系数线性微分方程主导李雅普诺夫指数的可微性，推导了导数公式并开发了数值算法，将其应用于平衡态种群的突变体入侵分析。
AI 中文摘要
对随时间波动环境中演化的研究通常依赖于李雅普诺夫指数的分析，该指数可量化种群的指数增长。然而，当模型参数依赖于随机过程时，计算作为表型性状演化预测关键工具的选择梯度就成为一项数学挑战。尽管周期环境下的情况已得到解决，但随机环境下的通用方法仍有待开发。本文提出一种严格方法，用于：1）建立主导李雅普诺夫指数关于某一参数的可微性，给出其导数的显式积分公式；2）开发一种数值算法，通过求解扩展微分方程来近似该导数；3）将上述结果应用于平衡态种群中突变体入侵的分析，确定选择梯度即主导李雅普诺夫指数的导数。
英文摘要
The study of evolution in temporally fluctuating environments often relies on the analysis of Lyapunov exponents, which quantify the exponential growth of populations. However, when model parameters depend on a stochastic process, calculating the selection gradient, a key tool for predicting the evolution of phenotypic traits, becomes a mathematical challenge. While the periodic case has been resolved, a general approach for random environments remains to be developed. This article proposes a rigorous method to: Establish the differentiability of the leading Lyapunov exponent with respect to a parameter, providing an explicit integral formula for its derivative. Develop a numerical algorithm to approximate the derivative by solving an extended differential equation. Apply these results to the analysis of mutant invasion in a resident population at equilibrium, identifying the selection gradient as the derivative of the top Lyapunov exponent.
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