Electron. J. Diff. Eqns., Vol. 2008(2008), No. 104, pp. 1-36.

A general framework for stability analysis of gene regulatory networks with delay

Irina Shlykova, Arcady Ponosov, Yury Nepomnyashchikh, Andrei Shindiapin

Abstract:
A method to study asymptotic properties of solutions to systems of differential equations with distributed time-delays and Boolean-type nonlinearities (step functions) is offered. Such systems arise in many applications, but this paper deals with specific examples of such systems coming from genetic regulatory networks. A challenge is to analyze stable stationary points which belong to the discontinuity set of the system (thresholds). The paper describes an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. The basic technical tool consists in replacing step functions by special smooth functions ("the tempered nonlinearities") and investigating the systems thus obtained.

Submitted May 20, 2008. Published August 6, 2008.
Math Subject Classifications: 34K60, 92D10.
Key Words: Gene regulation; delay equations; stability.

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Irina Shlykova
CIGENE - Centre for Integrative Genetics,
Department of Mathematical Sciences and Technology
Norwegian University of Life Sciences, N-1430 As, Norway
email: irinsh@umb.no
Arcady Ponosov
CIGENE - Centre for Integrative Genetics
Department of Mathematical Sciences and Technology
Norwegian University of Life Sciences, N-1430 As, Norway
email: arkadi@umb.no
Yury Nepomnyashchikh
CIGENE - Centre for Integrative Genetics
Department of Mathematics and Informatics
Eduardo Mondlane University, C.P. 257 Maputo, Mozambique
email: yuvn2@yandex.ru
Andrei Shindiapin
CIGENE - Centre for Integrative Genetics
Department of Mathematics and Informatics
Eduardo Mondlane University C.P. 257 Maputo Mozambique
email: andrei.olga@tvcabo.co.mz

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