TRAVELING WAVE SOLUTION OF A CHEMOTAXIS MODEL WITH LOGISTIC GROWTH AND INDIRECT SIGNAL PRODUCTION

This work is concerned with a non-monotonic chemotaxis model consisting of three equations. We first prove the existence of global solutions and the global attractivity of the uniform coexistence state in a moving coordinate frame. Then, we convert the existence of traveling wave solutions to the existence of fixed points of the solution mapping of the appropriate auxiliary system. Next, the upper and lower solutions of the auxiliary system are constructed, and the invariant domain of the solution mapping is established, so that the Schauder's fixed point theorem can be used. Finally, the non-existence condition of traveling wave solution is derived by developing principal eigenvalue theory. The obtained results reveal the effects of chemotaxis, the growth rate and the environmental carrying capacity on population invasion behavior.