  Cooperative interactions pervade in a broad range of many-body populations,
such as ecological communities, social organizations, and economic webs. We
investigate the dynamics of a population of two equivalent species A and B that
are driven by cooperative and symmetric interactions between these species. For
an isolated population, we determine the probability to reach fixation, where
only one species remains, as a function of the initial concentrations of the
two species, as well as the time to reach fixation. The latter scales
exponentially with the population size. When members of each species migrate
into the population at rate $\lambda$ and replace a randomly selected
individual, surprisingly rich dynamics ensues. Ostensibly, the population
reaches a steady state, but the steady-state population distribution undergoes
a unimodal to trimodal transition as the migration rate decreases below a
critical value $\lambda_c$. In the low-migration regime, $\lambda<\lambda_c$,
the steady state is not truly steady, but instead strongly fluctuates between
near-fixation states, where the population consists of mostly A's or of mostly
B's. The characteristic time scale of these fluctuations diverges as
$\lambda^{-1}$. Thus in spite of the cooperative interaction, a typical
snapshot of the population will contain almost all A's or almost all B's.
