Analysis of the Hopf bifurcation for the family of angiogenesis models

Abstract

In this paper we study a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and dʼOnofrio–Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper depends on two time delays and a parameter α ∈ [ 0 , 1 ] which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different α ∈ [ 0 , 1 ] , especially for α = 1 and α = 0 which reflects the Hahnfeldt et al. and the dʼOnofrio–Gandolfi models. For comparison we use also the value α = 1 / 2 .