In a bounded smooth domain \(\Omega \subset {\mathbb {R}}^{2}\) and with a positive parameter \(\chi >0\), we consider the chemotaxis system

for modeling the interactions between tumor and immune cells. It is shown that for any given \(\chi >0\) and for suitably regular initial data \((u_0, v_0, w_0)\), the corresponding homogeneous Neumann initial-boundary problem admits a global classical solution that is bounded; moreover, a critical mass phenomenon was analytically observed: If \(\chi \) is appropriately small, then the solution will approach spatially constant nontrivial equilibria in the large time limit provided that \(\bar{u}_0:=|\Omega |^{-1}\int _\Omega u_0<1\), whereas the solution will converge to its large time limit \((\bar{ u}_0, 0, 0)\) in the case \(\bar{u}_0\ge 1\).

在有界光滑域\（\ Omega \ subset {\ mathbb {R}} ^ {2} \）中，并使用正参数\（\ chi> 0 \），我们考虑了趋化系统

用于模拟肿瘤和免疫细胞之间的相互作用。结果表明，对于任何给定的\（\ chi> 0 \）和合适的常规初始数据\（（u_0，v_0，w_0）\），相应的齐次Neumann初边界问题允许有界的全局经典解；此外，通过分析观察到一个临界质量现象：如果\（\ chi \）适当小，则只要\（\ bar {u} _0：= | \ Omega | ^ {-1} \ int _ \ Omega u_0 <1 \），而在\（\ bar的情况下，解决方案将收敛到其较大的时间限制\（（\ bar {u} _0，0，0）\）{u} _0 \ ge 1 \）。