Spatial Pattern of an Epidemic Model with Cross-diffusion

Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coefficients. However, combined with cross-diffusion, the system will show emergence of isolated groups, i.e., stripe-like or spotted or coexistence of both, which we show by both mathematical analysis and numerical simulations. Our study shows that the interaction of self- and cross-diffusion can be considered as an important mechanism for the appearance of complex spatiotemporal dynamics in epidemic models.     

Pattern dynamics of network-organized system with cross-diffusion

Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results.     

双层非线性耦合反应扩散系统中复杂Turing斑图

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图. 关键词： Brusselator模型 非线性耦合 Turing模     

局域浓度调控扩散系数的次氯酸-碘离子-丙二酸系统图灵斑图形成中的反常扩散

通过数值模拟及振幅方程解析解方法,从实空间和倒空间分析了受局域浓度扩散系数调控下次氯酸-碘离子-丙二酸反应扩散系统图灵斑图形成的扩散机理.在零扩散系数调节下,斑图形成为典型的菲克扩散;而在负向正向扩散系数调节下,斑图的形成依赖欠扩散和超扩散.图灵系统的浓度稳态振幅对随机初始条件敏感性随局域浓度扩散调控系数k的增大而增加.     

双层耦合非对称反应扩散系统中的超点阵斑图

通过线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中图灵模之间的相互作用以及斑图的形成机理.模拟结果表明,合适的波数比以及相同的对称性是两个图灵模之间达到空间共振的必要条件,而耦合强度则直接影响了图灵斑图的振幅大小.为了保证对称性相同,两个图灵模的本征值高度要位于一定的范围内.只有失稳模为长波模时,才能对另一个图灵模产生调制作用,并形成多尺度时空斑图.随着波数比的增加,短波模子系统依次经历黑眼斑图、白眼斑图以及时序振荡六边形斑图的转变.研究表明失稳图灵模与处于短波不稳定区域的高阶谐波模之间的共振是产生时序振荡六边形的主要原因.     

Spatial pattern formation of a ratio-dependent predator-prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.     

Pattern formation in a predator–prey model with spatial effect

In this paper, spatial dynamics in the Beddington–DeAngelis predator–prey model with self-diffusion and cross-diffusion is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. Numerical simulations show that this system exhibits complex dynamical behaviors. In the Turing space, we find three types of typical patterns. One is the coexistence of hexagon patterns and stripe patterns. The other two are hexagon patterns of different types. The obtained results well enrich the finding in predator–prey models with Beddington–DeAngelis functional response.     

Stochastic Simulation of Turing Patterns

We investigate the effects of intrinsic noise on Turing pattern formation near the onset of bifurcation from the homogeneous state to Turing pattern in the reaction-diffusion Brusselator. By performing stochastic simulations of the master equation and using Gillespie＇s algorithm, we check the spatiotemporal behaviour influenced by internal noises. We demonstrate that the patterns of occurrence frequency for the reaction and diffusion pro- cesses are also spatially ordered and temporally stable. Turing patterns are found to be robust against intrinsic fluctuations. Sfochastic simulations also reveal that under the influence of intrinsic noises, the onset of Turing instability is advanced in comparison to that predicted deterministically.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Turing instability for a two-dimensional Logistic coupled map lattice

In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space-time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion.     

Pattern formation in reaction-diffusion system in crossed electric and magnetic fields

We consider a reaction-diffusion system in crossed electric and magnetic fields lying on the reaction plane. It is shown that a charge separation along the direction normal to the reaction plane resulting in a diffusional flux may cause a differential flow induced chemical instability and stationary pattern formation on a homogeneous steady state. This pattern is generically different from a Turing pattern modified by the crossed fields. The special role of magnetic field is emphasized. Our theoretical analysis is corroborated by numerical simulation on a reaction-diffusion system in three dimensions.     

Chaotic Turing pattern formation in spatiotemporal systems

The problem of Turing pattern formation has attracted much attention in nonlinear science as well as physics, chemistry and biology. So far spatially ordered Turing patterns have been observed in stationary and oscillatory media only. In this paper we find that spatially ordered Turing patterns exist in chaotic extended systems. And chaotic Turing patterns are strikingly rich and surprisingly beautiful with their space structures. These findings are in sharp contrast with the intuition of pseudo-randomness of chaos. The richness and beauty of the chaotic Turing patterns are attributed to a large variety of symmetry properties realized by various types of self-organizations of partial chaos synchronizations.     

六边形格子态斑图的数值模拟

采用双层耦合的Lengel-Epstein模型, 通过改变两子系统图灵模的强度比, 获得了四种的六边形格子态和多种非格子态结构. 模拟结果表明: 反应扩散系统的格子态结构由三套子结构叠加而成, 是两图灵模的波数比和强度比共同作用的结果, 两模的强度比决定了三波共振的具体模式; 另外, 系统选择格子态斑图所需的两图灵模的强度比大于非格子态斑图的强度比; 逐步增加两图灵模强度比, 出现的斑图趋于从复杂到简单变化. 深入研究发现: 不同互质数对(a, b)对应的格子态斑图的稳定性不同, 其中(3, 2)对应的格子态结构最为稳定.     

Turing patterns beyond hexagons and stripes

The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.     

Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel-Epstein model. Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.     

Complex turing patterns in non-linearly coupled systems

Recently pattern formation in layered structures, showing complicated superimposed patterns, has been modeled by coupling two Turing systems linearly, i.e., passively, such that the characteristic length scales of the independent systems are well separated. Here we propose a model of two non-linearly coupled Turing systems to study pattern formation in layered membrane-like structures, where the coupling plays an active role and changes the kinetics of the uncoupled systems. Extensive numerical simulations show that non-linear coupling generates a number of new regular patterns different from the ones observed earlier with linearly coupled systems. Some of them turn out to be superimposed patterns with different length scales, but many are not. Also, contrary to the linear coupling case, the strength of the non-linear coupling is found to play an important role in the formation and selection of patterns.     

Effect of a Local Source or Sink of Inhibitor on Turing Patterns

We investigate a reaction-diffusion model in which a Turing pattern develops and reproduces the formation of periodic segments behind a propagating chemical wave front. The chemical scheme involves two species known as activator and inhibitor. The model can be used to mimic the formation of prevertebrae during the early development of vertebrate embryo. Deterministic and stochastic analyses of the reaction-diffusion processes are performed for two typical sets of parameter values, far from and close to the Turing bifurcation. The effects of a local source or sink of inhibitor on the growing structure are studied and successfully compared with experiments performed on chick embryos. We show that fluctuations may lead to the formation of additional prevertebra.     

Pattern selection in a predation model with self and cross diffusion

In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator--prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for μ₁<μ≤μ₂, the steady state is the only stable solution of the model; for μ₂<μ≤μ₄, by increasing the control parameter μ, the sequence H_π-hexagons → H₀-hexagon-stripe mixtures rightarrow stripes → H_π-hexagon-stripe mixtures → H₀-hexagons is observed; for μ>μ₄, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator--prey model.     

Spatial recurrence strategies reveal different routes to Turing pattern formation in chemical systems

We analyze the temporal evolution of hexagonal Turing patterns in two Belousov-Zhabotinsky reactions performed in water-in-oil reverse micro-emulsions under different experimental conditions. The two reactions show different routes to pattern formation through localized spots and through a self replication mechanism. The Generalized Recurrence Plot (GRP) and the Generalized Recurrence Quantification Analysis (GRQA) are used for the investigation of spatial patterns and clearly reveal the different routes leading to the formation of stationary Turing structures.