Negativity of delayed induced oscillations in a simple linear DDE

作者：Forys Urszula; Bodnar Marek^*; Poleszczuk Jan

来源：Applied Mathematics Letters, 2011, 24(6): 982-986.

DOI：10.1016/j.aml.2011.01.010

• Delay differential equations
• Asymptotic analysis
• Hopf bifurcation
• Negativity of solutions
• Biochemical reactions
• Genetic regulatory network

In this work we study oscillations appearing in the simple linear delayed differential equation (DDE) of the form (x) over dot = A - Bx(t) - Cx(t - tau) with B < C in the case of tau larger than the critical value tau(cr) for which Hopf bifurcation occurs. We study the Cauchy problem proposed by Bratsun et al. (PNAS 102 (41) (2005)) as a description of some channel of biochemical reactions, that is we assume that x(t) = 0 for t < 0 and x(0) = x(0) >= 0. We prove that for any B < C and tau >= tau(cr), there exists a t in the interval (0.4 tau) for which x loses positivity. We conclude that the proposed Cauchy problem is not a proper description of biochemical reactions or of other biological and physical quantities.
We also consider another Cauchy problem with constant positive initial data. There exists a large set of initial data for which the solution to such a problem becomes negative. Therefore, this Cauchy problem is not a proper description of biological or physical quantities.

• 出版日期2011-6