Critical mass of lymphocytes for the coexistence in a chemotaxis system modeling tumor–immune cell interactions

Abstract

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

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u -\chi \nabla \cdot (u \nabla v), \\ v_t=\Delta v +w-v-uv, \\ w_t=\Delta w-uw + w(1-w), \end{array} \right. \end{aligned}$$

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 $\bar{u}_0\ge 1$.