Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

Abstract

For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in (τ, ε) parameter plane (τ for time delay, ε for diffusion rate). It is revealed that joint effects of diffusion and delay can lead to the occurrence of mixed spatial and temporal patterns. Moreover, it is also demonstrated that various spatially inhomogeneous patterns with different spatial frequencies can be achieved via changing the diffusion rate. And, the phenomenon that time delay may induce a failure of Turing instability observed by Gaffney and Monk (2006) are theoretically explained.

Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

“…steady state bifurcation resulting from double zero eigenvalues occurs, which is called Turing-Turing bifurcation in this paper and [10]. Actually, it is never easy to discuss local structures and stabilities of these bifurcating solutions.…”

“…For this purpose, we employ the method in [10] to accomplish Turing bifurcation analysis of (1) on a two-dimensional domain, by selecting d and ε as bifurcation parameters. Firstly, we determine a feasible region R in d-ε parameter plane, in which Turing bifurcation might occur when bifurcation parameters d and ε are chosen.…”

“…Model (2) which describes spatial distribution of morphogen, has been served as a typical model for studying spatial pattern formations of reaction-diffusion equations. And, dynamical behaviors of (2) defined on one-dimensional domain have been extensively investigated from many aspects, like Turing instability [15,36], Hopf bifurcation [14,35], Turing-Hopf bifurcation [10,22], asymmetric spike patterns [30]. Recently, time delay in gene expression is also proposed to be incorporated into (2), and it turns out that time delay plays a significant role in dynamics of system (2), see [6,26,1].…”

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

“…By Theorem 2.7, L 1 is called the first Turing bifurcation curve and it is sufficiently smooth but not piecewise smooth, which is different from the one derived in[21].By combining the results of Lemmas 2.3 and 2.6, Theorem 2.4 and following the definitions above, we have the following results. Assume that assumptions (H) and (H 1 ) hold.…”

“…As far as we know, although the existence of spatiotemporal phenomena has been studied to much extent through the Turing-Hopf bifurcation in many population dynamical models [1,4,7,21,27,32,33], there are few or no results in Turing-Hopf bifurcations for the competition system. Thus investigating what cause the occurrence of the Turing-Hopf bifurcation and how the spatiotemporal pattern can be formed is meaningful and worth exploring.…”