Global well-posedness and asymptotic behaviour for a reaction–diffusion system of competition type

We analyse a reaction–diffusion system describing the growth of microbial species in a model of flocculation type that arises in biology. A generalized model is formulated on a one dimensional bounded domain with feed terms at one end of the interval. Existence of global classical positive solutions is proved under general growth assumptions, with polynomial flocculation and deflocculation rates 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 nonhomogeneous 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.