Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

Abstract

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

Diffusion‐driven codimension‐2 Turing–Hopf bifurcation in the general Brusselator model

“…Restricted on 𝜁 k , we know that (10) and (11) have the same eigenvalues. Hence, Λ is the only eigenvalue of J k (u * (0), v * (0)), which has eigenvalues with zero real part for some k ∈ N 0 .…”

“…Then, by the procedure developed in previous studies, [44][45][46] (10) 10 − g (10) 01 + ig (10) 01 ∕ √ 𝑓 (a * ), B 12 = g (01) 10 , B 31 = g (10) 10 − g (10) 01 + d 1 k * 2 ∕𝑓 (a * )𝓁 2 g (10) 01 , B 32 = g (01) 10 , the calculation of B 210 , B 102 , B 111 , and B 003 , we leave them in Appendix A1.…”

“…a * 𝑓 ′′ (a * ) 𝑓 (a * ) , g (10) 11 = g (00) 21 = 𝑓 ′′ (a * ), g (00) 30 = a * b * 𝑓 ′′ ′ (a * ) 𝑓 (a * ) .…”

“…In fact, the set of all eigenvalues of system (10) with zero real part is Λ = {±𝜔 c , 0}. Our strategy is to decompose X into the eigenvector spaces and consider (10) in each eigenvector space. From (4), we set 𝜁 k = span{[𝜑, 𝜙 (i) k ]𝜙 (i) k |𝜑 ∈ X, i = 1, 2}.…”

“…Recently, many Turing-type models described by coupled systems of reaction-diffusion equations have been extensively studied in chemical and biological contexts. [9][10][11][12][13] In this paper, we shall consider the following general Brusselator model:…”