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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


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:

first{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

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

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


  1. [1]  Boczko, E.M., Gedeon, T., Stowers, C.C., and Young, T.R. (2010), ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast, Journal of Biological Dynamics, 4, 328-345. DOI: 10.1080/17513750903288003.
  2. [2]  Young, T.R., Fernandez, B., Buckalew, R., Moses, G., and Boczko, E.M. (2012), Clustering in cell cycle dynamics with general response/signaling feedback, Journal of Theoretical Biology, 292, 103-115. DOI: 10.1016/j.jtbi.2011.10.002.
  3. [3]  Breitsch, N., Moses, G., Boczko, E.M., and Young, T.R. (2014), Cell cycle dynamics: clustering is universal in negative feedback systems, Journal of Mathematical Biology, 70(5), 1151-1175. DOI: 10.1007/s00285-014-0786-7.
  4. [4]  Buckalew, R. (2014), Cell Cycle Clustering in a Nonlinear Mediated Feedback Model, DCDS B, 19(4), 867-881.
  5. [5]  Gong, X., Moses, G., Neiman, A., and Young, T.R. (2014), Noise-induced dispersion and breakup of clusters in cell cycle dynamics, Journal of Theoretical Biology, 335, 160-169. DOI: j.jtbi.2014.03.034.
  6. [6]  Gong, X., Buckalew, R., Young, T.R., and Boczko, E. (2014), Cell cycle dynamics in a response/signaling feedback system with a gap, Journal of Biological Dynamics, 8(1), 79-98. DOI:10.1080/17513758.2014.904526.
  7. [7]  Morgan, L., Moses, G., and Young, T.R. (2018), Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating, Letters in Biomathematics, 5, 113-128. DOI: 10.1080/23737867.2018.1456366.
  8. [8]  B\`{a}r\`{a}ny, B., Moses, G., and Young, T.R. (2019), Instability of the Steady State Solution in Cell Cycle Population Structure Models with Feedback, Journal of Mathematical Biology, 78(5), 1365-1387. DOI: 10.1007/s00285-018-1312-0.
  9. [9]  Rombouts, J., Prathom, K. and Young, T.R. (2020), Clusters tend to be of equal size in a negative feedback population model of cell cycle dynamics, SIAM Journal of Applied Dynamical Systems, 19(2), 1540-1573. DOI: 10.1137/19M129070X.
  10. [10]  Young, T.R., Prathom, K., and Rombouts, J. (2019), Temporal clustering in cell cycle dynamics, Dynamical Systems Magazine,
  11. [11]  Moses, G. (2015), Dynamical systems in biological modeling: clustering in the cell division cycle of yeast, Dissertation, Ohio University,
  12. [12]  Prathom, K. (2019), Stability Regions of Cyclic Solutions under Negative Feedback and Uniqueness of Periodic Solutions for Uneven Cluster Systems, Dissertation, Ohio University,
  13. [13]  Diekmann, O., Heijmans, H., and Thieme, H. (1984), On the stability of the cell size distribution, Journal of Mathematical Biology, 19, 227-248. DOI:10.1007/BF00277748.
  14. [14]  Diekmann, O., Heijmans, H., and Thieme, H. (1993), Perturbing semigroups by solving Stieltjes renewal equations, Journal of Differential and Integral Equations, 6, 155-181.
  15. [15]  Hannsgen, K.B., Tyson, J.J., and Watson, L.T. (1985), Steady-state size distributions in probabilistic models of the cell division cycle, SIAM Journal of Applied Mathematics, 45(4), 523-540. DOI:10.1137/0145031.
  16. [16]  Heijmans, H.J.A.M. (1984), On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72, 19-50. DOI:10.1016/0025-5564(84)90059-2.
  17. [17]  Rabi, K.C. (2020), Study of Some Biologically Relevant Dynamical System Models: (In)stability Regions of Cyclic Solutions in Cell Cycle Population Structure Model Under Negative Feedback and Random Connectivities in Multi-type Neuronal Network Models Dissertation, Ohio University.
  18. [18]  Perko, L. (2000), Differential Equations and Dynamical Systems, Springer-Verlag, 3rd edition.
  19. [19]  Bose, N. (1989), Tests for Hurwitz and Schur properties of convex combination of complex polynomials, IEEE Transactions on Circuits and Systems, 36(9), 1245-1247. DOI: 10.1109/31.34672.
  20. [20]  Fell, H.J. (1980), On the zeros of convex combinations of polynomials, Pacific Journal of Mathematics, 89(1), 43-50.
  21. [21]  Lang, S. (2002), Algebra, Springer-Verlag.
  22. [22]  Rotman, J. and Cuoco, A. (2013), Learning Modern Algebra, Mathematical Association of America.
  23. [23]  Rabi, K.C., and Abdalnaser, A. Roots of staircase palindromic polynomials. Preprint, arxiv:2012.15663.