In this paper, we consider the initial-boundary value problem of the two-species chemotaxis Keller-Segel model
$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla w), &x\in \varOmega , \ t>0, \\ v_{t}=\Delta v-\chi_{2}\nabla \cdot (v\nabla w), &x\in \varOmega , \ t>0, \\ 0=\Delta w-\gamma w+\alpha_{1}u+\alpha_{2}v, &x\in \varOmega , \ t>0, \end{cases}\displaystyle \end{aligned}$$
where the parameters
\(\chi_{1}\),
\(\chi_{2}\),
\(\alpha_{1}\),
\(\alpha_{2}\),
\(\gamma \) are positive constants,
\(\varOmega \subset \mathbb{R}^{2}\) is a bounded domain with smooth boundary. We obtain the results for finite time blow-up and global bounded as follows: (1) For any fixed
\(x_{0}\in \varOmega \), if
\(\chi_{1}\alpha_{2}= \chi_{2}\alpha_{1}\),
\(\int_{\varOmega }(u_{0}+v_{0})|x-x_{0}|^{2}dx\) is sufficiently small, and
\(\int_{\varOmega }(u_{0}+v_{0})dx>\frac{8\pi ( \chi_{1}\alpha_{1}+\chi_{2}\alpha_{2})}{\chi_{1}\alpha_{1}\chi_{2} \alpha_{2}}\), then the nonradial solution of the two-species Keller-Segel model blows up in finite time. Moreover, if
\(\varOmega \) is a convex domain, we find a lower bound for the blow-up time; (2) If
\(\|u_{0}\|_{L^{1}(\varOmega )}\) and
\(\|v_{0}\|_{L^{1}( \varOmega )}\) lie below some thresholds, respectively, then the solution exists globally and remains bounded.