Journal of Applied Nonlinear Dynamics
Instability of $k$Cluster Solutions in a Cell Cycle Population Model when $k$ is Prime
Journal of Applied Nonlinear Dynamics 11(1) (2022) 87138  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 subtriangles on which the stability of all cyclic solutions
are the same.
The stability for subtriangles 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 subtriangles.
Interior subtriangles with the order of events called $\mathbf{sr1}$,
were shown to have unstable solutions.
In the present work, we focus on interior subtriangles with the other order of events $\mathbf{rs1}$.
We show that when $k$ is prime,
then the cyclic solutions are unstable for all interior subtriangles.
When $k$ is even, we show that there always exist a small number of subtriangles
on which cyclic solutions are at least neutrally stable. For $k$ odd and composite,
we show that there are stable subtriangles 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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