Stability of Hahnfeldt angiogenesis models with time lags

P. Amster, L. Berezansky, L. Idels

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Mathematical models of angiogenesis, pioneered by 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 delay differential systems with non-Lipschitz nonlinearities. 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.

Original languageEnglish
Pages (from-to)2052-2060
Number of pages9
JournalMathematical and Computer Modelling
Issue number9-10
StatePublished - 1 May 2012


  • Angiogenesis
  • Equilibria
  • Global and local stability
  • Lienard equations
  • M-matrix
  • Non-Lipschitz nonlinearities
  • Nonlinear nonautonomous delay differential equations

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computer Science Applications


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