Stability analysis of the family of tumour angiogenesis models with distributed time delays

•A family of angiogenesis models with distributed delays is considered.•Erlang and piecewise linear (shifted or not) delay distributions are considered.•Stability regions of the positive steady state are investigated.•Existences of the Hopf bifurcation phenomena are studied.•For the limit cases stability results for models with discrete delays are recovered.

In the present paper a family of angiogenesis models that is a generalisation of the Hahnfeldt et al. model is proposed. Considered family of models consists of two differential equations with distributed time delays. The global existence and the uniqueness of the solutions are proved. Moreover, the stability of the unique positive steady state is examined in the case of Erlang and piecewise linear delay distributions. Theorems guaranteeing the existence of stability switches and occurrence of Hopf bifurcations are proved. Theoretical results are illustrated by numerical analysis performed for parameters estimated by Hahnfeldt et al. (1999) [47].