This paper deals with a nonlinear system of two partial differential equations arising in chemotaxis, involving a source term of logistic type. The existence of global bounded classical solutions is proved under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough. Also, the existence of global weak solutions is shown under rather mild conditions. Secondly, the corresponding stationary problem is studied and some regularity properties are given. It is proved that in presence of certain, sufficiently strong logistic damping there is only one nonzero equilibrium, and all solutions of the non-stationary system approach this steady state for large times. On the other hand, for small logistic terms some multiplicity and bifurcation results are established.

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka–Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.