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

•We analyse the of family of models describing the angiogenesis process.•We examine the existence of Hopf bifurcation for DDE system with two unequal delays.•We analyze Hopf bifurcation type when one of delays is a bifurcation parameter.•Our analysis covers the case of well-know Hahnfeldt et al. model with included delays.•We extend the results from the previous work where only one delay is nonzero.

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.