Pinero, Jordi; S. Redner and Ricard Sole

Cooperative interactions pervade the dynamics of a broad rage of many-body systems, such as ecological communities, the organization of social structures, and economic webs. In this work, we investigate the dynamics of a simple population model that is driven by cooperative and symmetric interactions between two species. We develop a mean-field and a stochastic description for this cooperative two-species reaction scheme. For an isolated population, we determine the probability to reach a state of fixation, where only one species survives, as a function of the initial concentrations of the two species. We also determine the time to reach the fixation state. When each species can migrate into the population and replace a randomly selected individual, the population reaches a steady state. We show that this steady-state distribution undergoes a unimodal to trimodal transition as the migration rate is decreased beyond a critical value. In this low-migration regime, the steady state is not truly steady, but instead fluctuates strongly between near-fixation states of the two species. The characteristic time scale of these fluctuations diverges as λ−1.