TURING PATTERN FORMATION IN A TWO-SPECIES NEGATIVE FEEDBACK SYSTEM WITH CROSS-DIFFUSION

2013 ◽  
Vol 23 (09) ◽  
pp. 1350162 ◽  
Author(s):  
XIYAN YANG ◽  
ZHANJIANG YUAN ◽  
TIANSHOU ZHOU
Keyword(s):  
Pattern Formation ◽  
Negative Feedback ◽  
Sufficient Conditions ◽  
Feedback System ◽  
Turing Instability ◽  
Natural World ◽  
Mutual Diffusion ◽  
Turing Pattern ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion

Pattern formation is a ubiquitous phenomenon in the natural world. Previous studies showed that for an activator–inhibitor system without cross-diffusion, spatial patterns can be formed only when the diffusion of the inhibitor is significantly faster than that of the activator. However, cross-diffusion exists extensively in real systems, especially in biological systems. Here, we study a prototypic two-species negative feedback system with cross-diffusion. By performing stability analysis of equilibrium state, we find sufficient conditions for Turing instability. Both analytical and numerical results demonstrate that mutual diffusions of the two species can lead to the Turing pattern formation regardless of differences in self-diffusion coefficients. However, in the absence of the mutual diffusion or even if there is the cross-diffusion of only one species, the system cannot exhibit Turing patterns. Our results reveal the mechanism of Turing pattern formation in a class of reaction–diffusion systems, where mutual diffusion between species plays a key role.

Turing Pattern Formation from the Cooperation of Competition and Cross-Diffusion

2014 ◽  
Vol 24 (03) ◽  
pp. 1450038 ◽  
Author(s):  
Xiyan Yang ◽  
Hongwei Yin ◽  
Tianshou Zhou
Keyword(s):  
Pattern Formation ◽  
Negative Feedback ◽  
Reaction Diffusion ◽  
Feedback Loops ◽  
Turing Patterns ◽  
Turing Pattern ◽  
Cross Diffusion ◽  
Interacting Species ◽  
Negative Feedback Loops ◽  
Positive And Negative Feedback

This paper investigates the pattern formation in a reaction–diffusion (R-D) system where two interacting species form coupled positive and negative feedback loops. It is found that the cooperation of competition and cross-diffusion can lead to the Turing pattern formation for which an adequate set of conditions are analytically derived. Such a mechanism of generating Turing patterns is different from the case that self-diffusion is sufficient to generate Turing patterns in a paradigm model (proverbially called as the Turing model) where two interacting species constitute a single negative feedback loop. Therefore, this work not only provides another model paradigm for studying the pattern formation but also would be helpful for understanding the formation of, for example, diversiform skin patterns in the mammalian world where coupled positive and negative feedback loops are ubiquitous.

scholarly journals Pattern Formation in a Cross-Diffusive Holling-Tanner Model

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Weiming Wang ◽  
Zhengguang Guo ◽  
R. K. Upadhyay ◽  
Yezhi Lin
Keyword(s):  
Pattern Formation ◽  
Sufficient Conditions ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
Flux Boundary Conditions ◽  
Spatially Distributed

We present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals Turing pattern formation in a predator–prey system with cross diffusion

2014 ◽  
Vol 38 (21-22) ◽  
pp. 5022-5032 ◽  
Author(s):  
Zhi Ling ◽  
Lai Zhang ◽  
Zhigui Lin
Keyword(s):  
Pattern Formation ◽  
Turing Pattern ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System
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