For the following two-species chemotaxis system with two signal productions: 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{lll} u_t=\Delta u-\chi \nabla \cdot \left( u\nabla v\right) , & x \in \Omega , t>0, \\ 0=\Delta v-\mu _w+w, & x \in \Omega , t>0, \\ w_t=\Delta w-\xi \nabla \cdot \left( w\nabla z\right) , & x \in \Omega , t>0,\\ 0=\Delta z-\mu _u+u, & x \in \Omega , t>0, \end{array} \end{array}\right. } \end{aligned}\) under homogeneous Neumann boundary conditions in a smooth, bounded domain \(\Omega \subset \mathbb {R}^2\) , where \(\mu _u=-\!\int _\Omega u(\cdot ,0)\) and \(\mu _w=-\!\int _\Omega w(\cdot ,0)\) . We establish new criteria distinguishing between finite-time blow-up and boundedness for radially symmetric solutions, that are if \(\Omega =B_R(0)\subset \mathbb {R}^2\) , then for any initial datum \((u_0,w_0)\) which is radially symmetric and more mass-concentrated than the spatially homogeneous average also satisfies \(\begin{aligned} m_u:=\int _\Omega u_0\geqslant \frac{32\pi }{\xi },\quad m_w:=\int _\Omega w_0\geqslant \frac{32\pi }{\chi }, \end{aligned}\) the corresponding solution blows up in finite time;
if \(\Omega =B_R(0)\subset \mathbb {R}^2\) , then for any initial datum \((u_0,w_0)\) which is radially symmetric and satisfies \(\begin{aligned} m_u< \frac{8\pi }{\xi },\quad m_w< \frac{8\pi }{\chi }, \end{aligned}\) the corresponding solution remains uniformly bounded.
