Yan Guo, Hyung Ju Hwang
We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the finite number of linear growing modes over a time scale of $ln(1/δ)$, where &δ$ is the strength of the initial perturbation.
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Turing Instability Suppressed and Induced by Multiplicative Noise in Brusselator System
该研究探讨乘性噪声对Brusselator系统Turing不稳定性的作用，发现同强度双浓度噪声可抑制不稳定、单浓度噪声可诱发不稳定，并通过线性化量化噪声幅值、数值模拟验证结论。
In this paper, we study a strongly coupled two-prey one-predator system. We first prove the unique positive equilibrium solution is globally asymptotically stable for the corresponding kinetic system (the system without diffusion) and remains locally linearly stable for the reaction-diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. Moreover we prove that the positive equilibrium solution is globally asymptotically stable for the reaction-diffusion system without cross-diffusion. But it becomes linear unstable only when cross-diffusio…
The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalised spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing type instabilities. These results are numerica…
Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction-diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators i…
The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diff…
Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilib…
We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focused on the model of early carcinogenesis proposed by Marciniak-Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non-diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns due to diffusi…
Turing instability analysis of a singular cross-diffusion problem
针对Busenberg和Travis奇异种群模型需正则化求解的问题，本文通过弱非线性稳定性分析发现正则化问题的无界波数斑图在极限问题中因振荡衰减不存在，并用有限元模拟验证了结论。
Turing patterns in reaction-diffusion (RD) systems have classically been studied only in RD systems which do not explicitly depend on independent variables such as space. In practise, many systems for which Turing patterning is important are not homogeneous with ideal boundary conditions. In heterogeneous systems with stable steady states, the steady states are also necessarily heterogeneous which is problematic for applying the classical analysis. Whilst there has been some work done to extend Turing analysis to some heterogeneous systems, for many systems it is still difficult to determine i…
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary…
Pattern formation and nonlinear waves close to a 1:1 resonant Turing and Turing--Hopf instability
研究1:1空间共振下Turing与Turing–Hopf不稳定性附近的模式形成，推导耦合复Ginzburg–Landau振幅方程并构造周期解、快速锋面及全局空间周期解。
Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach
本文提出一种计算机辅助证明方法，通过基变换和Gershgorin圆盘定理处理异质非局部反应扩散系统中的强耦合无限微分系统，证明了图灵不稳定性，并得出分岔点的精确阈值。
Turing chemotactic instability in an HIV model
本文在含靶细胞logistic增长的HIV反应-扩散模型中引入趋化作用，发现更低的图灵失稳阈值，从而更好地解释早期感染热点形成。
Nakao and Mikhailov proposed using continuous models (mean-field models) to study reaction-diffusion systems on networks and the corresponding Turing patterns. This work aims to show that p-adic analysis is the natural tool to carry out this program. This is possible due to the fact that the discrete Laplacian attached to a network turns out to be a particular case of a p-adic Laplacian. By embedding the graph attached to the system into the field of p-adic numbers, we construct a continuous p-adic version of a reaction-diffusion system on a network. The existence and uniqueness of the Cauchy …
We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove no…
Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard-Marangoni problem
本文针对三维Bénard-Marangoni问题在Turing不稳定性附近的渐近长波模型，推导振幅方程并利用中心流形理论研究了方形与六边形图案的分岔，构造了描述图案间过渡的快速移动调制前沿解，为实验观察到的图案形成提供了可能的理论机制。
We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from classical modulation theory can be adapted for use in the fast-slow setting, by reformulating it as a blow-up transformation. This leads to dynamically simpler modulation equations posed in the blown-up space, via a formal procedure which directly extends the established approach to the time-dependent setting. The modulation equations take the form of non-au…
We study a p-adic reaction-diffusion system and the associated Turing patterns. We establish an instability criteria and show that the Turing patterns are not classical patterns consisting of alternating domains. Instead of this, a Turing pattern consists of several domains (clusters), each of them supporting a different pattern but with the same parameter values. This type of patterns are typically produced by reaction-diffusion equations on large networks.
Quantum Turing Patterns
该研究在林德布拉德晶格动力学中构建量子图灵斑图，建立其严格理论，揭示量子纠缠与波长选择的关联，数值模拟呈现多种斑图形态。
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