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.
Related
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- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
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- Backward stochastic partial differential equations with unbounded random coefficients
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BibTeX
@misc{carmona2026differentiability,
title = {{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}},
author = {Philippe Carmona},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.02711},
eprint = {2609.02711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02711},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - 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
AU - Philippe Carmona
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.02711
UR - https://arxiv.org/abs/2609.02711
AB - 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.
ER -