Turing Patterns in the Lengyel–Epstein System with Superdiffusion

2017 ◽  
Vol 27 (08) ◽  
pp. 1730026 ◽  
Author(s):  
Biao Liu ◽  
Ranchao Wu ◽  
Naveed Iqbal ◽  
Liping Chen
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Complex Dynamics ◽  
Multiple Scale ◽  
Turing Instability ◽  
Amplitude Equations ◽  
Homogeneous Steady State ◽  
Weakly Nonlinear ◽  
Homogeneous Solutions ◽  
Normal Diffusion

Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

scholarly journals Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Feifan Zhang ◽  
Wenjiao Zhou ◽  
Lei Yao ◽  
Xuanwen Wu ◽  
Huayong Zhang
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Analysis ◽  
Discrete Model ◽  
Continuous Model ◽  
Spatiotemporal Patterns ◽  
Homogeneous Steady State ◽  
Simulation Results

In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750073 ◽  
Author(s):  
Peng Feng
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Allee Effect ◽  
Turing Instability ◽  
Spatial Pattern Formation ◽  
The Stability ◽  
Steady State Solutions

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

Turing Patterns of a Lotka–Volterra Competitive System with Nonlocal Delay

2018 ◽  
Vol 28 (07) ◽  
pp. 1830021 ◽  
Author(s):  
Bang-Sheng Han ◽  
Zhi-Cheng Wang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Turing Patterns ◽  
Amplitude Equations ◽  
Competitive System ◽  
Scale Method ◽  
Nonlocal Delay

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

scholarly journals Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

2021 ◽  
Vol 28 (4) ◽  
pp. 363-389
Author(s):  
Shao-Yue Mi ◽  
Bang-Sheng Han ◽  
Yu-Tong Zhao
Keyword(s):  
Multiple Scale ◽  
Turing Instability ◽  
Turing Patterns ◽  
Scale Analysis ◽  
Cooperative System ◽  
Amplitude Equations ◽  
Pattern Dynamics ◽  
Pattern Structures ◽  
New Phenomena ◽  
Multiple Scale Analysis

AbstractThis paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

Bifurcation, Chaos and Turing Instability Analysis for a Space-Time Discrete Toxic Phytoplankton-Zooplankton Model with Self-Diffusion

2019 ◽  
Vol 29 (13) ◽  
pp. 1950184
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
You Li ◽  
Nan Jiang
Keyword(s):  
Numerical Simulations ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Space Time ◽  
Discrete Version ◽  
Flip Bifurcation ◽  
Center Manifold Theorem ◽  
Toxic Phytoplankton ◽  
Stable Conditions

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

CROSS-DIFFUSION INDUCED INSTABILITY IN LVLEV–TANNER MODEL

2011 ◽  
Vol 04 (04) ◽  
pp. 431-442 ◽  
Author(s):  
WENZHEN GAN ◽  
PENG ZHU ◽  
JIE BAO
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Asymptotically Stable ◽  
Homogeneous Steady State ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Ode System ◽  
Uniform Steady State ◽  
Parameter Values

A Lvlev–Tanner model with cross-diffusion is considered. We analyze the positive uniform steady state and obtain conditions on the parameter values such that the homogeneous steady state is locally asymptotically stable both in the related ODE system and in the PDE system with self-diffusion. Once also cross-diffusion is considered in the model, the uniform steady state is shown to be unstable under some conditions. Numerical simulations are also presented.

scholarly journals On finite-amplitude patterns of convection in a rectangular-planform container

1996 ◽  
Vol 317 ◽  
pp. 111-127 ◽  
Author(s):  
P. G. Daniels ◽  
M. Weinstein
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Numerical Simulations ◽  
Steady State Solution ◽  
Critical Value ◽  
Finite Amplitude ◽  
State Structure ◽  
Amplitude Equations ◽  
Critical Wavelength ◽  
Rayleigh Numbers

This paper considers the development of finite-amplitude patterns of convection in rectangular-planform containers. The horizontal dimensions of the container are assumed to be large compared with the critical wavelength of the motion. An interaction between rolls parallel and perpendicular to the lateral boundaries is modelled by a coupled pair of nonlinear amplitude equations together with appropriate conditions on the four lateral boundaries. At Rayleigh numbers above a critical value a steady-state solution is established with rolls parallel to the shorter lateral sides, consistent with the predictions of linear theory. At a second critical value this solution becomes unstable to cross-rolls near the shorter sides and a new steady state evolves. This consists of the primary roll pattern together with regions near the shorter sides where there is a combination of rolls parallel and perpendicular to the boundary.Analytical and numerical methods are used to describe both the evolution and steady-state structure of the solution, and a comparison is made with the results of full numerical simulations and experiments.

scholarly journals Spatiotemporal patterns induced by four mechanisms in a tussock sedge model with discrete time and space variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
You Li ◽  
Jingjing Cao ◽  
Ying Sun ◽  
Dan Song ◽  
Xiaoyu Wu
Keyword(s):  
Numerical Simulations ◽  
Spatial Patterns ◽  
Discrete Time ◽  
Turing Instability ◽  
Spatiotemporal Patterns ◽  
Spatial Diffusion ◽  
Time And Space ◽  
Homogeneous Solutions ◽  
Kinetic System

AbstractIn this paper, we investigate the spatiotemporal patterns of a freshwater tussock sedge model with discrete time and space variables. We first analyze the kinetic system and show the parametric conditions for flip and Neimark–Sacker bifurcations respectively. With spatial diffusion, we then show that the obtained stable homogeneous solutions can experience Turing instability under certain conditions. Through numerical simulations, we find periodic doubling cascade, periodic window, invariant cycles, chaotic behaviors, and some interesting spatial patterns, which are induced by four mechanisms: pure-Turing instability, flip-Turing instability, Neimark–Sacker–Turing instability, and chaos.

A Consistent Implementation of Inelastic Material Models into Steady State Rolling

2016 ◽  
Vol 44 (3) ◽  
pp. 174-190 ◽  
Author(s):  
Mario A. Garcia ◽  
Michael Kaliske ◽  
Jin Wang ◽  
Grama Bhashyam
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Viscoelastic Material ◽  
Rolling Contact ◽  
Arbitrary Lagrangian Eulerian ◽  
Ale Formulation ◽  
Inelastic Material ◽  
Material Formulation ◽  
Steady State Rolling ◽  
Efficient Alternative

ABSTRACT Rolling contact is an important aspect in tire design, and reliable numerical simulations are required in order to improve the tire layout, performance, and safety. This includes the consideration of as many significant characteristics of the materials as possible. An example is found in the nonlinear and inelastic properties of the rubber compounds. For numerical simulations of tires, steady state rolling is an efficient alternative to standard transient analyses, and this work makes use of an Arbitrary Lagrangian Eulerian (ALE) formulation for the computation of the inertia contribution. Since the reference configuration is neither attached to the material nor fixed in space, handling history variables of inelastic materials becomes a complex task. A standard viscoelastic material approach is implemented. In the inelastic steady state rolling case, one location in the cross-section depends on all material locations on its circumferential ring. A consistent linearization is formulated taking into account the contribution of all finite elements connected in the hoop direction. As an outcome of this approach, the number of nonzero values in the general stiffness matrix increases, producing a more populated matrix that has to be solved. This implementation is done in the commercial finite element code ANSYS. Numerical results confirm the reliability and capabilities of the linearization for the steady state viscoelastic material formulation. A discussion on the results obtained, important remarks, and an outlook on further research conclude this work.

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

Sign in / Sign up

Export Citation Format

Share Document