Phase transition from poor to diverse ecosystems

A mathematical model of ecoevolution is studied. The model treats ecosystems as large dimensional dynamical systems. The preying interaction term between species have the scale invariant form of . In addition, simple rules for addition and elimination of species are included. This model is called the “scale-invariant” model. The model makes it possible to construct ecosystems with thousands of species with a totally random invasion process, although it is not impossible when the interaction terms are the quadratic form of xixj like Lotka–Volterra equation. We studied the relation between the number of species and the interspecies interactions. As a result, it is shown the model can describe both simple ecosystems and diverse ecosystems, because this model has two phases. In one phase, the number of species remains in finite range. In the other phase, the number of species grows without limit.