Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals

Summary

We study a nonlinear system of differential equations describing the evolution of a competitive two-species chemotaxis system with two chemicals in a bounded domain. The system consists of four PDEs, two equations of parabolic type describing the evolution of the competitive species and two elliptic equations modeling the distribution of the chemicals. By introducing global competitive/cooperative factors, we obtain, for different ranges of parameters, that any positive and bounded solution converges to a spatially homogeneous state. The proofs rely on the comparison principle for spatially homogeneous sub- and super-solutions. The existence and uniqueness of global classical solution are proved under assumptions on the initial data and appropriate conditions on the parameters of the system. Such solution stabilizes to spatially homogeneous equilibria in the large time limit, given coexistence or extinction of solutions for a range of parameters.