  Many-variable differential equations with random coefficients provide
powerful models for the dynamics of many interacting species in ecology. These
models are known to exhibit a dynamical phase transition from a phase where
population sizes reach a fixed point, to a phase where they fluctuate
indefinitely. Here we provide a theory for the critical behavior close to the
phase transition. We show that timescales diverge at the transition and that
temporal fluctuations grow continuously upon crossing it. We further show the
existence of three different universality classes, with different sets of
critical exponents, depending on the migration rate which couples the system to
its surroundings.
