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The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

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  • A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

    Mathematics Subject Classification: Primary: 35B32, 37G15, 35B36, 92C15; Secondary: 37L10.


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  • Figure 1.  A: The parameter space of $(a, b)$ which satisfies Theorem 2.4. In the white region, we will have a stable periodic solution for the FDEs. B: A bifurcation diagram for $\tau$ for the FDEs. This has been plotted using the value of $u(t)$ at which $v(t)=v_*$. The parameter values $(a, b)=(0.1, 0.9)$ have been chosen and the system exhibits the first Hopf bifurcation at approximately $\tau_0=0.2171$. For $\tau\in[0, \tau_0)$, $E_*=(1.0, 0.9)$ is always asymptotically stable, bifurcating first at $\tau_0$ then exhibiting subsequent bifurcations with frequency doubling prior to chaotic dynamics

    Figure 2.  A homogeneous periodic solution for $u$ with $0< \varepsilon<1$. The parameter values for the simulations are given by $a=0.1, b=0.9, \varepsilon=0.1$. A: A plot of $u$ for $d=0.5$, when equation (43) is satisfied. B: A plot of $u$ for $d=5\times 10^{-5}$, when equation (43) is not satisfied

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