In this paper, we consider an angiogenesis free boundary tumor model with external periodic nutrient supply. The model is in the form of a reaction–diffusion equation describing the concentration of nutrients $\sigma$ and an elliptic equation describing the distribution of the internal pressure $p$. The vasculature provides a periodic supply of nutrients to the tumor at a rate proportional to $\beta$, so that $\frac{\partial \sigma }{\partial \mathbf{n}}+\beta (\sigma -\varphi \left(t\right))=0$ holds on the boundary, where $\varphi \left(t\right)$ is the nutrient concentration outside the tumor. Here $\varphi \left(t\right)$ is a periodic function with period $T$ and satisfies $\varphi \left(t\right)=\varphi (t+T)$. A parameter $\mu$ in the model expresses the “aggressiveness” of the tumor. We prove that under non-radially symmetric perturbations, there exists a ${\mu }_{\ast }>0$ such that the $T$-periodic solution is linearly stable for $\mu <{\mu }_{\ast }$, and is linearly unstable for $\mu >{\mu }_{\ast }$.

在本文中，我们考虑具有外部周期性营养供应的无血管生成边界肿瘤模型。该模型采用反应扩散方程式的形式描述营养素的浓度$\sigma$ 和描述内部压力分布的椭圆方程 $p$。脉管系统以与肿瘤成比例的速率为肿瘤提供周期性的营养$\beta$， 以便 $\frac{\partial \sigma }{\partial \mathbf{ñ}}+\beta （\sigma -\varphi （Ť））=0$ 保持在边界上 $\varphi （Ť）$是肿瘤外的营养物浓度。这里$\varphi （Ť）$ 是周期的周期函数 $Ť$ 并满足 $\varphi （Ť）=\varphi （Ť+Ť）$。一个参数$\mu$在模型中表达了肿瘤的“侵略性”。我们证明在非径向对称扰动下，存在一个${\mu }_{\ast }>0$ 这样 $Ť$周期解对于 $\mu <{\mu }_{\ast }$，并且对于 $\mu >{\mu }_{\ast }$。