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Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence. (English) Zbl 0886.92020

The authors analyze a linear model of cell population dynamics structured by age (denoted a) with two interacting compartments: proliferating cells (with densities \(p(a,t))\) and quiescent cells (with densities \(q(a,t))\); \(t\) is time. The equations of the model are: \[ \partial p/ \partial t+\partial p/ \partial a= -\mu(a)p -\sigma(a)p +\tau (a)q, \quad 0<a <a_1,\;t>0, \]
\[ \partial q/ \partial t+\partial q/ \partial a= \sigma(a)p -\tau (a)q, \quad 0<a< a_1,\;t>0, \]
\[ p(0,t)= 2\int^q_0 \mu(a) p(a,t)da, \quad t>0,\quad q(0,t)=0,\;t>0, \]
\[ p(a,0)= \varphi (a),\quad 0<a<a_1,\quad q(a,0) =\psi(a),\quad 0<a<a_1, \] where \(\mu\) is the division rate, \(\sigma\) is the transition rate from the proliferating stage to the quiescent stage, \(\tau\) is the transition rate from the quiescent stage to the proliferating stage, and \(a_1\) is maximal age of division.
Necessary and sufficient conditions are established for the population to exhibit asymptotic behavior of asynchronous exponential growth. The model is analyzed as a semigroup of linear operators.

MSC:

92D25 Population dynamics (general)
47N60 Applications of operator theory in chemistry and life sciences
47D03 Groups and semigroups of linear operators
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