Global Behavior in a Two-Species Chemotaxis-Competition System with Signal-Dependent Sensitivities and Nonlinear Productions

This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions {=del center dot((1)()del-(1)()del)+(1)(1-(1)-(1)), is an element of Omega, >0, = Delta- + (1)(1), is an element of Omega, >0, =del center dot((2)()del-(2)()del)+(2)(1-2-(2)), is an element of Omega, >0, =Delta-+(2)(2), is an element of Omega, >0 in a bounded and smooth domain Omega subset of(2), where the parameters ,,,, (=1,2) are positive constants, and the functions (1)(), (1)(), (2)(), (2)() fulfill the following hypotheses: lozenge (()), ()is an element of (2)([0,infinity)), (), ()>0 for all >= 0, '() < 0 and lim(->infinity)()=0; lozenge lim(->infinity) ()/() and lim(->infinity) '()/() exist. We first confirm the global boundedness of the classical solution provided that the additional conditions 2(1) <= 1+(2) and 2(2) <= 1+(1) hold. Moreover, by constructing several suitable Lyapunov functionals, it is demonstrated that the global solution exponentially or algebraically converges to the constant stationary solutions and the corresponding convergence rates are determined under some specific stress conditions.