  We analyze a reaction-diffusion system describing the growth of microbial
species in a model of flocculation type that arises in biology. Existence of
global classical positive solutions is proved under general growth assumptions,
with flocculation and deflocculation rates polynomially bounded above, that
guarantee uniform sup norm bounds for all time t obtained by an $L^{p}-$energy
functional estimate. We also show finite time blow up can occur when the yield
coefficients are large enough. Also, using arguments relying on the spectral
and fixed theory, we show persistence and existence of nonhomogenous population
steady-states. Finally, we present some numerical simulations to show the
combined effects of motility coefficients and the flocculation-deflocculation
rates on the coexistence of species.
