Emergence of Chaos, Diffusive Chaos and Strange Attractors in Evolving Volterra Ecosystem

This paper begins with an introduction to the emergence of chaos in a game of evolution proposed recently (Dubois, 1998). The law of conservation of materials in nutrients and populations is used as an environmental closure. Malthusian growth is so transformed to a Pearl-Verhulst map. The game of evolution deals with the competition between a species with its successive mutants. Such a population with random mutations evolves when the ratio birth rateldeath rate of a mutant increases. Chaos appears in such an evolving ecosystem.

In this paper, several new basic models of nutrients and population interaction are presented and simulated.

Firstly, a second order Pearl-Verhulst is proposed : a second time derivative term is added to the classical Pear-Verhulst model. This term permits to control the velocity of propagation of a population by spatial diffusion. With low value of the diffusion coefficient, the population front is followed by a spatial uniform concentration of the population. For higher values of the diffusion coeflicient bifircations then chaos appear in the spatial structure of the population. This is what we already called a "diffusive chaos" (Dubois, 1996, 1998).

Secondly, this second order Pearl-Verhulst can show either the classical chaos either a strange attractor similar to Hénon's attractor (1976). The final states in the bifurcation depends on the initial conditions : this system has a memory of its initial conditions, and the system goes to different attraction basins.

Thirdly, the nutrients N - population P interaction model is complicated in adding an intermediate state P* for the population : P* is the satiated population and only non satiated population P can take nutrients. Surprisingly, such an ecosystem has memory but also anticipatory properties similar to the incursive model of the Pearl-Verhulst given before (Dubois, 1996). Such a system depends on the initial conditions and show a strange attractor similar to the Hénon attractor.