Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays

TL;DR

This paper analyzes a free boundary model of tumor growth incorporating angiogenesis and two delays, demonstrating conditions for Hopf bifurcation and exploring how delays influence tumor dynamics through numerical simulations.

Contribution

It extends previous work by establishing Hopf bifurcation conditions in a tumor growth model with delays and angiogenesis.

Findings

Hopf bifurcation occurs under specific conditions.

Delays significantly affect tumor stability and oscillations.

Numerical simulations illustrate the relationship between delays and bifurcation.

Abstract

This paper concerns a free boundary problem modeling tumor growth with angiogenesis and two time delays. The two delays represent the time taken for cells to undergo mitosis and modify the rate of cell loss because of apoptosis, respectively. We study the stability of stationary solutions and find that Hopf bifurcation occurs under some conditions, which extends the results of Xu. Furthermore, numerical simulations are performed to investigate the relationship among the rate of angiogenesis, two time delays and Hopf bifurcation.