Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Abstract

We considered the differential geometric structure of non-equilibrium dynamics in non-linear interactions, such as competition and predation, based on Kosambi-Cartan-Chern (KCC) theory. The stability of a geodesic flow on a Finslerian manifold is characterized by the deviation curvature (the second invariant in the dynamical system). According to KCC theory, the value of the deviation curvature is constant around the equilibrium point. However, in the non-equilibrium region, not only the value but also the sign of the deviation curvature depend on time. Next, we reapplied KCC theory to the dynamics of the deviation curvature and determined the hierarchical structure of the geometric stability. The dynamics of the deviation curvature in the nonequilibrium region is accompanied by a complex periodic (node) pattern in the predation (competition) system.