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

TL;DR

This paper analyzes a multi-layer tumor growth model with periodic external nutrients, establishing conditions for the stability of zero or positive solutions based on nutrient averages, and considers a flat tumor shape.

Contribution

It provides a complete classification of the stability of solutions in a multi-layer tumor model with periodic nutrients, including cases with non-spherical tumor shapes.

Findings

Zero equilibrium is globally stable if average nutrients are below threshold.

Positive periodic solution exists and is stable if average nutrients exceed threshold.

The tumor shape considered is flat, differing from typical spherical models.

Abstract

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 $\Phi(t)$ and threshold concentration for proliferation $\widetilde{\sigma}$. In this paper, we give a complete classification about $\Phi(t)$ and $\widetilde{\sigma}$ 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 $\frac{1}{T} \int_{0}^{T} \Phi(t)d t>\widetilde{\sigma}$, then there exists a unique positive T-periodic solution and it is globally stable.