Global and local stability for a non-linear hyperbolic system model for the role of stem cells in physiological homeosta
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Global and local stability for a non-linear hyperbolic system model for the role of stem cells in physiological homeostasis Laura Di Bernardo · Donatella Donatelli
Received: 22 September 2014 / Revised: 29 December 2014 / Accepted: 30 December 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract In this paper we propose an existence and uniqueness theory for the solutions of a system of non-linear hyperbolic conservation laws, structured in age and maturity variables, representing a tissue environment. In particular we are interested in the investigation of the role of stem cells in its homeostasis. The main result presented in this paper is the consistence of the stability results arising from the analysis of the model we designed with the experimental observations on which several branches of medicine are currently attempting to trade on their research activity. Keywords
Age-dependent · Gompertzian growth · Stem cells
Mathematics Subject Classification
35F20 · 35B35 · 92C50
1 Introduction A good population model has to reproduce satisfactorily behaviors observed by Biologists. A significant class of models which are able to fulfill this requirements are the structured one. They describe the distribution of individuals through different classes, determined by individual differences related to decisive factors of the dynamics we are interested in. This type of models have the advantage to be able to make a connection between the individual and the population level so that they are able to catch dynamical behaviors which other models can not.
Communicated by Pierangelo Marcati. L. Di Bernardo · D. Donatelli (B) DISIM, Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, 67100 L’Aquila, AQ, Italy e-mail: [email protected] L. Di Bernardo e-mail: [email protected]
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L. Di Bernardo, D. Donatelli
The first authors that attempted to construct a structured model, as a general frame able to describe the intrinsic non linearity of real biological systems, where Gurtin and MacCamy in Gurtin and MacCamy (1974). Their work, afterwards, has inspired the rise of a wide literature; see Ani¸ta et al. (1998), Iannelli (1994), Hoppensteadt (1975), Marcati (1982), Marcati (1983), and Webb (1985). In this paper we design a model able to confirm the experimental evidence that stem cells are able to maintain and eventually recover the homeostasis of both animal and vegetal tissues. In order to do this, our system has been outlined in three groups of cells cohabiting and co-existing, has follows: – Stem cells that are undifferentiated cells able both to self-convert into specialized units or just to divide depending on the external signals. – Proliferating cells that are the healthy building blocks of the tissue as well as the headquarters of its specific physiological functions. – Damaged cells which are cells that, due to an error in their protein synthesis traffic, caused by alterations in their genetic patrimony, show an i
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