Negativity of delayed induced oscillations in a simple linear DDE

Abstract

In this work we study oscillations appearing in the simple linear delayed differential equation (DDE) of the form x ̇ = A − B x ( t ) − C x ( t − τ ) with B < C in the case of τ larger than the critical value τ 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 τ ≥ τ cr there exists a t in the interval ( 0 , 4 τ ) 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.