  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 $\lambda^{-1}$.
