Gompertz model with delays and treatment: Mathematical analysis

Marek Bodnar; Monika Joanna Piotrowska; Urszula Foryś

doi:10.3934/mbe.2013.10.551

Article Contents

2013, Volume 10, Issue 3: 551-563. Doi: 10.3934/mbe.2013.10.551

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Gompertz model with delays and treatment: Mathematical analysis

1.
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw

Received: May 31, 2012

Revised: July 31, 2012

Published: March 2013

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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.

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Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 34K20, 34K28, 37N25; Secondary: 92B05, 92B25, 92C50.

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