Analysis of the Hopf bifurcation for the family of angiogenesis models II: The case of two nonzero unequal delays

Abstract

In this paper we continue the analysis of 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 of models 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. Previously, in Piotrowska and Foryś (2011) [11] we have considered three cases with either one of the delays equals to 0 or both delays equal to each other. Here, we focus on the case with two unequal and non-zero delays present in the model, and study the dynamics depending on the parameter α, including stability switches, Hopf bifurcation and stability of arising periodic orbits for different α∈[0,1], especially for α=1 and α=0 which reflects the Hahnfeldt et al. model and d’Onofrio & Gandolfi model, respectively. Moreover, we consider the influence of constant treatment on the model dynamics. It occurs that the treatment not only decreases the tumour size at a steady state but also enlarges the region of stability.