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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:

p/t+p/a=-μ(a)p-σ(a)p+τ(a)q,0<a<a 1 ,t>0,
q/t+q/a=σ(a)p-τ(a)q,0<a<a 1 ,t>0,
p(0,t)=2 0 q μ(a)p(a,t)da,t>0,q(0,t)=0,t>0,
p(a,0)=ϕ(a),0<a<a 1 ,q(a,0)=ψ(a),0<a<a 1 ,

where μ is the division rate, σ is the transition rate from the proliferating stage to the quiescent stage, τ 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:
92D25Population dynamics (general)
47N60Applications of operator theory in biology and other sciences
47D03(Semi)groups of linear operators