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)

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

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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