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Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Instability of $k$-Cluster Solutions in a Cell Cycle Population Model when $k$ is Prime

Journal of Applied Nonlinear Dynamics 11(1) (2022) 87--138 | DOI:10.5890/JAND.2022.03.007

Rabi K.C., Abdalnaser Algoud, Todd R.Young

Department of Mathematics, Ohio University, Athens, OH 45701, USA

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Abstract

We study periodic `cyclic' solutions of a population model of the cell cycle. In this model, clusters of cells in one fixed phase of the cell cycle may exert a negative influence on the progress of clusters in another phase. Previous results showed that stability of cyclic solutions is determined by the values of model parameters $s$ and $r$, and by which of two possible orderings of certain events the cyclic solution follows. The parameter triangle $\triangle = \{(s,r): 0 \le s \le r \le 1 \}$ is subdivided into sub-triangles on which the stability of all cyclic solutions are the same. The stability for sub-triangles on the boundary of $\triangle$ was fully characterized in terms of number theoretic relationships between the number of clusters $k$ and certain indices of the sub-triangles. Interior sub-triangles with the order of events called $\mathbf{sr1}$, were shown to have unstable solutions. In the present work, we focus on interior sub-triangles with the other order of events $\mathbf{rs1}$. We show that when $k$ is prime, then the cyclic solutions are unstable for all interior sub-triangles. When $k$ is even, we show that there always exist a small number of sub-triangles on which cyclic solutions are at least neutrally stable. For $k$ odd and composite, we show that there are stable sub-triangles when $k = 9$ and $k=15$ and no others.

Acknowledgments

The authors thank Saad Aldosari, Herath Mudiyanselage Indupama Herath and Daniel Ntiamoah for many long, tedious and helpful discussions.

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