Steady States Analysis of a Nonlinear Age-Structured Tumor Cell Population Model with Quiescence and Bidirectional Trans
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Steady States Analysis of a Nonlinear Age-Structured Tumor Cell Population Model with Quiescence and Bidirectional Transition Zijian Liu1
· Chunfang Guo2 · Jin Yang1 · Hong Li3
Received: 2 April 2019 / Accepted: 27 December 2019 © Springer Nature B.V. 2020
Abstract A nonlinear age-structured tumor cell population model with quiescence and bidirectional transition is presented and studied. The division rate of the proliferating cells is assumed to be nonlinear due to the limitation of the nutrient and space. The model includes a proportion of newborn cells that enter directly the quiescent phase with age zero. This proportion can reflect the effect of treatment by drugs such as erlotinib. The bidirectional transition between proliferating cells and quiescent cells is considered. The local and global stabilities of the trivial steady state are investigated. The existence and local stability of the positive steady state are also analyzed. Numerical simulations are performed to verify the results and to examine the impacts of parameters on the nonlinear dynamics of the model. Keywords Age-structured model · Proliferating and quiescent stages · Bidirectional transition · Steady state analysis
1 Introduction Researchers of population dynamics not only focus on the variation of population number to the time, but also interest in the age distributions of many populations. Almost all of populations with life cycle can be analyzed theoretically or numerically by establishing models of age-structured forms. A variety of age structured growth models such as demographic models [1, 2], epidemic models [3–8], microscopic virus models [9–11] and cell population models [12–16] are presented. Meanwhile, many interesting mathematical properties including existence and uniqueness of solutions, existence and stability of steady states,
B Z. Liu
[email protected]
1
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R. China
2
School of Economics and Management, Beijing Jiaotong University, Beijing 100044, P.R. China
3
School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China
Z. Liu et al.
asynchronous exponential growth or ergodicity, periodic oscillation and bifurcations and so on are studied [17–23]. In [16], Dyson et al. investigated an age-structured cell population model, in which individual cells are distinguished as proliferating and quiescent stages and cells transition is allowed between these two stages at any age. They also allowed newly divided cells to enter quiescence with age zero. The authors proved that the model also admits an asynchronous exponential growth, which has been proved in [15] where the case that newly divided cells can enter into quiescent stage with age zero is not considered. A characteristic of asynchronous exponential growth in an age-structured population is that the total population grows exponentially, but the age-structure stabilizes in the sense that the fraction of cells in any age r
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