In this paper, we investigate the following two-species chemotaxis-competition system on \(\mathbb {R}^N\) 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi _1\nabla \cdot (u\nabla w)+\mu _1u(1-u-a_1v), & t>0,~x\in \mathbb {R}^N,\\ v_t=\Delta v-\chi _2\nabla \cdot (v\nabla w)+\mu _2v(1-v-a_2u), & t>0,~x\in \mathbb {R}^N, \\ w_t=\Delta w-w+u+v, & t>0,~x\in \mathbb {R}^N,\\ u(0,x)=u_0(x),~v(0,x)=v_0(x),~w(0,x)=w_0(x),& x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}\) where \(N\ge 1\) is a positive integer, \(\chi _i,\mu _i,a_i (i=1,2)\) are positive constants. We prove that system (0.1) has a unique global classical solution for every nonnegative, bounded, and uniformly continuous functions \(u_0(x)\) and \(v_0(x)\) , and every nonnegative, bounded, uniformly continuous, and differentiable function \(w_0(x).\) Moreover, we obtain the asymptotic behavior of classical solution of (0.1) in three competition cases. We mainly prove that suitably small chemotaxis sensitivity coefficients \(\chi _1,\chi _2\) are sufficient to determine the asymptotic behavior of classical solution of (0.1) in three competition cases.
