Gradient-driven dynamics on Finsler manifolds: The Jacobi action-metric theorem and an application in ecology

Abstract

Jacobi’s theorem on Riemannian action-metrics for gradient driven dynamics is generalized to Finsler manifolds for regular Finsler metric functions defined on positively-conical regions of
the slit tangent bundle. Both the stability of the full gradient dynamics and that of the geodesic flow of the action-metric are determined via KCC-theory and Finsler geodesic deviation. Calculations are carried out for 2-dimensional Berwald spaces with non-vanishing curvature using the software package,
FINSLER. These metrics are useful for models in ecology and evolution.