## International Journal of Differential Equations

### Linear Analysis of an Integro-Differential Delay Equation Model

Anael Verdugo

#### Abstract

This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of $N$ delay differential equations (DDEs) coupled to $N$ ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as $N$ becomes large.

#### Article information

Source
Int. J. Differ. Equ., Volume 2018 (2018), Article ID 5035402, 6 pages.

Dates
Accepted: 14 March 2018
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.ijde/1528855325

Digital Object Identifier
doi:10.1155/2018/5035402

Mathematical Reviews number (MathSciNet)
MR3801852

Zentralblatt MATH identifier
06915955

#### Citation

Verdugo, Anael. Linear Analysis of an Integro-Differential Delay Equation Model. Int. J. Differ. Equ. 2018 (2018), Article ID 5035402, 6 pages. doi:10.1155/2018/5035402. https://projecteuclid.org/euclid.ijde/1528855325

#### References

• I. Palmeirim, D. Henrique, D. Ish-Horowicz, and O. Pourquié, “Avian hairy gene expression identifies a molecular clock linked to vertebrate segmentation and somitogenesis,” Cell, vol. 91, no. 5, pp. 639–648, 1997.
• M. B. Elowitz and S. Leibier, “A synthetic oscillatory network of transcriptional regulators,” Nature, vol. 403, no. 6767, pp. 335–338, 2000.
• D. Bratsun, D. Volfson, L. S. Tsimring, and J. Hasty, “Delay-induced stochastic oscillations in gene regulation,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 102, no. 41, pp. 14593–14598, 2005.
• J. Lewis, “Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator,” Current Biology, vol. 13, no. 16, pp. 1398–1408, 2003.
• N. A. M. Monk, “Oscillatory expression of Hes1, p53, and NF-$\kappa$B driven by transcriptional time delays,” Current Biology, vol. 13, no. 16, pp. 1409–1413, 2003.
• A. Verdugo and R. Rand, “Hopf bifurcation in a DDE model of gene expression,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 2, pp. 235–242, 2008.MR2360687
• H. de Jong, “Modeling and simulation of genetic regulatory systems: a literature review,” Journal of Computational Biology, vol. 9, no. 1, pp. 67–103, 2002.
• T. Schlitt and A. Brazma, “Current approaches to gene regulatory network modelling,” BMC Bioinformatics, vol. 8, no. 6, article no. S9, 2007.
• H. De Jong, J.-L. Gouzé, C. Hernandez, M. Page, T. Sari, and J. Geiselmann, “Qualitative simulation of genetic regulatory networks using piecewise-linear models,” The Bulletin of Mathematical Biology, vol. 66, no. 2, pp. 301–340, 2004.MR2254590
• A. Mochizuki, “Structure of regulatory networks and diversity of gene expression patterns,” Journal of Theoretical Biology, vol. 250, no. 2, pp. 307–321, 2008.
• S. Turner, J. A. Sherratt, K. J. Painter, and N. J. Savill, “From a discrete to a continuous model of biological cell movement,” Physical Review E, vol. 69, 2004.
• J. Goutsias and S. Kim, “Stochastic transcriptional regulatory systems with time delays: A mean-field approximation,” Journal of Computational Biology, vol. 13, no. 5, pp. 1049–1076, 2006.
• A. Ribeiro, R. Zhu, and S. A. Kauffman, “A general modeling strategy for gene regulatory networks with stochastic dynamics,” Journal of Computational Biology, vol. 13, no. 9, pp. 1630–1639, 2006.
• R. Edwards, P. Van Den Driessche, and L. Wang, “Periodicity in piecewise-linear switching networks with delay,” Journal of Mathematical Biology, vol. 55, no. 2, pp. 271–298, 2007.
• A. Verdugo and R. H. Rand, “DDE Model of Gene Expression: A Continuum Approach,” in ASME Proceedings of IMECE, pp. 1112–1120, 2008. \endinput