A simple model of carcinogenic mutations with time delay and diffusion
In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely fav...
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| Format: | Article |
| Language: | English |
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AIMS Press
2013-03-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.861 |
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| author | Monika Joanna Piotrowska Urszula Foryś Marek Bodnar Jan Poleszczuk |
| author_facet | Monika Joanna Piotrowska Urszula Foryś Marek Bodnar Jan Poleszczuk |
| author_sort | Monika Joanna Piotrowska |
| collection | DOAJ |
| description | In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy.Included delay stands for the interactions between benign and other cells.We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists.The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion. |
| format | Article |
| id | doaj-art-06ce0df53d834544b61f5da8de457691 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2013-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-06ce0df53d834544b61f5da8de4576912025-01-24T02:26:12ZengAIMS PressMathematical Biosciences and Engineering1551-00182013-03-0110386187210.3934/mbe.2013.10.861A simple model of carcinogenic mutations with time delay and diffusionMonika Joanna Piotrowska0Urszula Foryś1Marek Bodnar2Jan Poleszczuk3Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawInstitute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawInstitute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawInstitute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 WarsawIn the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy.Included delay stands for the interactions between benign and other cells.We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists.The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.861delay equationsstabilitycarcinogenic mutations.diffusion |
| spellingShingle | Monika Joanna Piotrowska Urszula Foryś Marek Bodnar Jan Poleszczuk A simple model of carcinogenic mutations with time delay and diffusion Mathematical Biosciences and Engineering delay equations stability carcinogenic mutations. diffusion |
| title | A simple model of carcinogenic mutations with time delay and diffusion |
| title_full | A simple model of carcinogenic mutations with time delay and diffusion |
| title_fullStr | A simple model of carcinogenic mutations with time delay and diffusion |
| title_full_unstemmed | A simple model of carcinogenic mutations with time delay and diffusion |
| title_short | A simple model of carcinogenic mutations with time delay and diffusion |
| title_sort | simple model of carcinogenic mutations with time delay and diffusion |
| topic | delay equations stability carcinogenic mutations. diffusion |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.861 |
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