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Mathematical Biosciences and Engineering (MBE)
 

Gompertz model with delays and treatment: Mathematical analysis

Pages: 551 - 563, Volume 10, Issue 3, June 2013      doi:10.3934/mbe.2013.10.551

 
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Marek Bodnar - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)
Monika Joanna Piotrowska - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)
Urszula Foryƛ - Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email)

Abstract: In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.

Keywords:  Gompertz model, delay equation, Hopf bifurcation, stability, stability switches.
Mathematics Subject Classification:  Primary: 34K11, 34K13, 34K18, 34K20, 34K28, 37N25; Secondary: 92B05, 92B25, 92C50.

Received: June 2012;      Accepted: August 2012;      Available Online: April 2013.

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