The existence of periodic solution and asymptotic behavior of solutions for a multi-layer tumor model with a periodic provision of external nutrients

In this paper, we consider a multi-layer tumor model with a periodic provision of external nutrients. The domain occupied by tumor has a different shape (flat shape) than spherical shape which has been studied widely. The important parameters are periodic external nutrients $\mathrm{\Phi }(t)$ and threshold concentration for proliferation $\tilde{\sigma }$. In this paper, we give a complete classification about $\mathrm{\Phi }(t)$ and $\tilde{\sigma }$ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, if $\frac{1}{T} \int_{0}^{T} \Phi(t)d t\leqslant\widetilde{\sigma}$, then the zero equilibrium solution is globally stable while if $\frac{1}{T}{\int }_0^T\mathrm{\Phi }(t)dt>\tilde{\sigma }$, then there exists a unique positive T-periodic solution and it is globally stable.