Lotka Volterra Coevolution at the Edge of Chaos

In this paper, we study the coevolution of species by combining a theoretical approach with a computer simulation in order to show how a discrete distribution of viable species emerges. Coevolution is modelled as a replicator system which, with an additional diffusion term representing the mutation, leads to a Schrodinger equation. This system dynamics can be interpreted as a survival race between species on a multimodal sinking and drifting landscape whose modes correspond to the eigen modes of the Schrodinger equation. This coevolution dynamics is further illustrated by a simulation based on a continuous phenotypic model due to Kaneko in which the interactions between species are interpreted through a Lotka-Volterra model. This simulated coevolution is seen to converge to viable species associated with a dynamics at the edge of chaos (i.e with a null Lyapounov exponent). The transition from such a viable species to another results from some kind of tunnel effects characteristic of the punctuated equilibrium classically observed in biology in which rapid changes in the species distribution follow long plateaus of stable distribution.