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Predicting the evolution of expanding population is critical to controlbiological threats such as invasive species and virus explosion. In this paper,we consider a two species chemotaxis system of parabolic-parabolic-elliptictype with Lotka-Volterra type competition terms and a free boundary. Such amodel with a free boundary describes the spreading of new or invasive speciessubject to the influence of some chemical substances in an environment with afree boundary representing the spreading front. We first find conditions on theparameters which guarantee the global existence and boundedness of classicalsolutions with nonnegative initial functions. Next, we investigatevanishing-spreading dichotomy scenarios for positive solutions. It is shownthat the vanishing-spreading dichotomy in the generalized sense always occurs;that the vanishing spreading dichotomy in the strong sense occurs when thecompetition between two species is weak-weak competition; and that thevanishing spreading dichotomy in the weak sense occurs when the competitionbetween two species is weak-strong competition.

Vanishing-spreading dichotomy in a two-species chemotaxis competition system with a free boundary