Lotka–Volterra system and KCC theory: Differential geometric structure of competitions and predations

Abstract

We consider the differential geometric structure of competitions and predations in the sense of the Lotka–Volterra system based on KCC theory. For this, we visualise the relationship between the Jacobi stability and the linear stability as a single diagram. We find the following. (I) Ecological interactions such as competition and predation can be described by the deviation curvature. In this case, the sign of the deviation curvature depends on the type of interaction, which reflects the equilibrium point type. (II) The geometric quantities in KCC theory can be expressed in terms of the mean and Gaussian curvatures of the potential surface. In this particular case, the deviation curvature can be interpreted as the Willmore energy density of the potential surface. (III) When the equations of the system have nonsymmetric structure for the species (e.g. a predation system), each species also has nonsymmetric geometric structure in the nonequilibrium region, but symmetric structure around the equilibrium point. These findings suggest that KCC theory is useful to establish the geometrisation of ecological interactions.