Dynamics of a system for migration from proliferative to dormant status
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Dynamics of a system for migration from proliferative to dormant status Cory L. Howk
Received: 26 July 2014 / Revised: 17 December 2014 / Accepted: 26 December 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract The behavior of a mathematical system for a population with two classifications, proliferative and dormant, is explored. The system exhibits a one-dimensional manifold of nonisolated steady states. Phase plane trajectories are calculated and analyzed, and a convergent power series solution is derived, illustrating the complexity inherent when even a basic nonlinearity is introduced into an ODE system. An application of this fundamental system toward the differentiation of neural progenitor cells is then presented, illustrating close agreement with the results of the full ODE system for the physical process. Keywords Population dynamics · Nonlinear ODEs · Phase plane analysis · Nonisolated steady states · Bell polynomials · Adult hippocampal progenitor cells (AHPC) Mathematics Subject Classification
92D25 · 34A05 · 34D20 · 37N25
1 Introduction The mathematical modeling of interacting populations has a rich history dating back to the early twentieth century. The essential techniques of compartmentalization, mass action, functional response, parameter estimation, and stability/sensitivity analysis (among many others) have been used to study interacting species in ecology (Aranda et al. 2012; Bratbak and Thingstad 1985; Busenburg et al. 1990; Dwyer et al. 1997; Elderd et al. 2013; Esteva and Yang 2005; Freedman and Waltman 1977; Hadeler and Freedman 1989; Lewis and Murray 1993; Mills and Getz 1996; White et al. 1996), interacting cell populations in vivo (Collier et al. 1996; Dillon and Othmer 1999; Graham et al. 2013; Levine et al. 2000, 2001,
Communicated by Florence Hubert. C. L. Howk (B) Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723, USA e-mail: [email protected]
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2006; Bailon-Plaza and van der Meulen 2001; Reyser et al. 2010; Anderson and Chaplain 1998; Anderson 2005), even interacting chemical/molecular species (Field and Noyes 1974; Gombert and Nielsen 2000; Michaelis and Menten 1913; Nagy-Ungvarai et al. 1989; Pallson and Lightfoot 1984; Rizzi et al. 1997; Tzafriri 2003; Vaseghi et al. 1999). The same techniques have also been applied to the dynamics among multiple classifications of an individual species, such as disease transmission (Allen and Victory 2003; Derouich et al. 2003; Dietz and Hadeler 1988; Feng and Velasco-Hernandez 1997; Hethcote 2000; Jensen et al. 2006; Kermack and McKendrick 1927; Mohler et al. 2005; Perelson and Nelson 1999; Shulgin et al. 1998; Smith and Cheeseman 2002; Zhang et al. 2007), aging populations (Gyllenberg 1982; McCauley et al. 1993; Murphy and Smith 1990; Ronce et al. 2000; Schenzle 1984; Sinko and Streifer 1969; Thompson and Cauley 1979), and cell proliferation in vivo (Crauste et al. 2008; Dingli et al. 2007; Duomic et al. 2011; Johnston et al. 2007; Loef
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