Instability of k-Cluster Solutions in a Cell Cycle Population Model when k is Prime - K.C., Rabi; Algoud, Abdalnaser; Young, Todd R. (Journal of Applied Nonlinear Dynamics, 2026)
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 Δ = {(s,r): 0 s r 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 Δ 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.