Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

Abstract

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

Delay-induced Turing instability in reaction-diffusion equations

“…Furthermore, from Refs. [11,12] we conclude that, if E * is stable when τ = 0, then for τ c given above (a) the equilibrium E * is stable for all 0 τ < τ c , (b) E * is unstable for all τ > τ c , and (c) if Re( dλ k dτ )| τ =τ c > 0, Hopf bifurcation occurs at τ = τ c .…”

“…The bifurcation direction can be determined by using the normal form [11]. If we choose the time delay as the control parameter, then from this conclusion, we obtain the so-called Turing space [15][16][17], α > 0, 0 < γ < 1, β 0 < β < β 1 , and τ > τ c in which the Turing instability occurs.…”

“…In ecology, time delay is commonly used to measure a maturation, gestation period, or reaction time of a predator population [1,4,5], and mathematically it may result in a much richer dynamics for a system, such as inducing the instability of an equilibrium so that Hopf bifurcation occurs [6][7][8][9][10][11][12]. Therefore, systems with time delays have received wide attention over the past years.…”