Mathematical analysis of a model of chemotaxis with competition terms. (English) Zbl 1363.35041

Summary: We consider a competitive system of differential equations describing the behavior of two biological species “u” and “v”. The system is weakly coupled and one of the species has the capacity to diffuse and moves toward the higher concentration of the second species following its gradient, the density function satisfies a second order parabolic equation with chemotactic terms. The second species does not have motility capacity and satisfies an ordinary differential equation. We prove that the solutions are uniformly bounded and exist globally in time. The asymptotic behavior of solutions is also studied for a range of parameters and initial data. If the chemotaxis coefficient \(\chi\) is small enough the quadratic terms drive the solutions to the constant steady state.


35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
92D40 Ecology