Differentiability of the Leading Lyapunov Exponent of a linear differential equation with random coefficients Application to the Calculation of the Selection Gradient in Random Environments

Abstract: 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.