Stability of Hahnfeldt Angiogenesis Models with Time Lags

TL;DR

This paper investigates the stability of Hahnfeldt angiogenesis models incorporating biologically motivated time delays, providing new conditions for global and local stability and revealing complex dynamics.

Contribution

Introduces biologically motivated time-varying delays into angiogenesis models and derives explicit stability conditions using M-matrix theory.

Findings

Models with delays exhibit complex, nontrivial dynamics.

New stability criteria for positive solutions are established.

Local stability of one model is proven using recent Lienard-type delay results.

Abstract

Mathematical models of angiogenesis, pioneered by P. Hahnfeldt, are under study. To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M-matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies.