This paper is devoted to the fully parabolic two-species chemotaxis-competition system with nonlocal terms \(\begin{aligned} \left\{ \begin{aligned}&u_t=d_1\Delta u-\chi _1\nabla \cdot (u\nabla {w})+u\left( a_0-a_1u-a_2v-a_3 \int _{\Omega } u -a_4\int _{\Omega } v \right) , & x\in \Omega ,t>0,\\&v_t=d_2\Delta {v}-\chi _2\nabla \cdot (v\nabla {w})+v\left( b_0-b_1u-b_2v-b_3 \int _{\Omega } u -b_4\int _{\Omega } v \right) , & x\in \Omega ,t>0,\\&w_t=d_3\Delta {w}-\lambda {w}+\kappa u+\iota v, & x\in \Omega ,t>0, \end{aligned}\right. \end{aligned}\) under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\Omega \subseteq \mathbb {R}^n\) , \(n=1,2\) . We prove that this problem possesses a global classical solution which is uniformly bounded under the condition \((a_1+a_3|\Omega |)>0,~(b_2+b_4|\Omega |)>0,~ a_4 b_3 \le (a_1+a_3|\Omega |)(b_2+b_4|\Omega |)|\Omega |^{-2}\) in the case \(n\le 2\) . We obtain the fundamental estimate on \(\Vert u\Vert _{L^{1}(\Omega )}\) and \(\Vert v\Vert _{L^{1}(\Omega )}\) by deriving a subtle estimate for \(\int _{\Omega }(u+\rho ^2v)\) with \(\rho \) to be suitably chosen, which extends and improves the result of Xu (2020).
