Instability of $k$-Cluster Solutions in a Cell Cycle Population Model when $k$ is Prime
DOI:
https://doi.org/10.5890/JAND.2022.03.007Abstract
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 $\Delta = \{(s,r): 0 \le s \le r \le 1\}$ 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 $\Delta$ 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 sr1, were shown to have unstable solutions. In the present work, we focus on interior sub-triangles with the other order of events 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.References
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