Analysis of the Hopf bifurcation for the family of angiogenesis models |
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Authors: | Monika J Piotrowska Urszula Fory? |
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Institution: | Department of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland |
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Abstract: | In this paper we study 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 includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and d?Onofrio-Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper 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. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different α∈0,1], especially for α=1 and α=0 which reflects the Hahnfeldt et al. and the d?Onofrio-Gandolfi models. For comparison we use also the value α=1/2. |
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Keywords: | Angiogenesis Hopf bifurcation Stability Stability switches |
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