Stability of Hahnfeldt angiogenesis models with time lags

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• 作者：P. Amster^a ; ^{pamster@dm.uba.ar} ; L. Berezansky^b ; ^{brznsky@cs.bgu.ac.il} ; L. Idels^c ; ^{lev.idels@viu.ca}
• 关键词：Angiogenesis ; Nonlinear nonautonomous delay differential equations ; Global and local stability ; Equilibria ; Non-Lipschitz nonlinearities ; M-matrix ; Lienard equations
• 刊名：Mathematical and Computer Modelling
• 出版年：2012
• 出版时间：May, 2012
• 年：2012
• 卷：55
• 期：9-10
• 页码：2052-2060
• 全文大小：240 K

文摘

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 -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.