scholarly journals Emergence of Turing Patterns in a Simple Cellular Automata-Like Model via Exchange of Integer Values between Adjacent Cells

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Takeshi Ishida
Keyword(s):  
Cellular Automata ◽  
Turing Instability ◽  
Information Networks ◽  
Turing Patterns ◽  
Chemical Agent ◽  
Spatial Structures ◽  
Data Traffic ◽  
Control Data ◽  
Self Organized ◽  
Turing Model

The Turing pattern model is one of the theories used to describe organism formation patterns. Using this model, self-organized patterns emerge due to differences in the concentrations of activators and inhibitors. Here a cellular automata (CA)-like model was constructed wherein the Turing patterns emerged via the exchange of integer values between adjacent cells. In this simple hexagonal grid model, each cell state changed according to information exchanged from the six adjacent cells. The distinguishing characteristic of this model is that it presents a different pattern formation mechanism using only one kind of token, such as a chemical agent that ages via spatial diffusion. Using this CA-like model, various Turing-like patterns (spots or stripes) emerge when changing two of four parameters. This model has the ability to support Turing instability that propagates in the neighborhood space; global patterns are observed to spread from locally limited patterns. This model is not a substitute for a conventional Turing model but rather is a simplified Turing model. Using this model, it is possible to control the formation of multiple robots into such forms as circle groups or dividing a circle group into two groups, for example. In the field of information networks, the presented model could be applied to groups of Internet-of-Things devices to create macroscopic spatial structures to control data traffic.

scholarly journals Exploring Spatiotemporal Complexity of a Predator-Prey System with Migration and Diffusion by a Three-Chain Coupled Map Lattice

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Xuebing Cong ◽  
Ge Pan ◽  
Xiumin Zhang ◽  
...  
Keyword(s):  
Turing Instability ◽  
Coupled Map Lattice ◽  
Self Organization ◽  
Turing Patterns ◽  
Predator Prey ◽  
Migration Rates ◽  
Self Organized ◽  
Prey System ◽  
And Migration ◽  
And Diffusion

The topic of utilizing coupled map lattice to investigate complex spatiotemporal dynamics has attracted a lot of interest. For exploring the spatiotemporal complexity of a predator-prey system with migration and diffusion, a new three-chain coupled map lattice model is developed in this research. Based on Turing instability analysis, pattern formation conditions for the predator-prey system are derived. Via numerical simulation, rich Turing patterns are found with subtle self-organized structures under diffusion-driven and migration-driven mechanisms. With the variation of migration rates, the predator-prey system exhibits a gradual dynamical transition from diffusion-driven patterns to migration-driven patterns. Moreover, new results, the self-organization of non-Turing patterns, are also revealed. We find that even in the cases where the nonspatial predator-prey system reaches collapse, the migration can still drive pattern self-organization. These non-Turing patterns suggest many new possible ways for the coexistence of predator and prey in space, under the effects of migration and diffusion.

scholarly journals The Design Principles of Discrete Turing Patterning Systems

2020 ◽  
Author(s):  
Tom Leyshon ◽  
Elisa Tonello ◽  
David Schnoerr ◽  
Heike Siebert ◽  
Michael P.H. Stumpf
Keyword(s):  
Cellular Automata ◽  
Large Scale ◽  
Turing Instability ◽  
Turing Patterns ◽  
Biochemical Processes ◽  
Modelling Framework ◽  
Underlying Principle ◽  
Continuous Models ◽  
Turing Instabilities ◽  
Definition Of

AbstractThe formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-node networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find the Turing topologies to be substantially more robust to changes in the parameters of the model. We also find that Turing instabilities are a much weaker predictor for emerging patterns in simulations in our discrete modelling framework. We propose a modification of the definition of a Turing instability for cellular automata models as a better predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

scholarly journals STABILITY ANALYSIS OF CELLULAR AUTOMATA GENERATED CLUSTERS

2005 ◽  
Vol 12 (1) ◽  
pp. 83-90
Author(s):  
R. Šiugždaite
Keyword(s):  
Cellular Automata ◽  
Power Law ◽  
State Parameter ◽  
Urban System ◽  
Complex Behaviour ◽  
Cellular Automata Model ◽  
Self Organized ◽  
Tai Ji ◽  
Long Time ◽  
Ca Model

The development of regional urban system still remains one of the main problems during the human race history. There are a lot of problems inside this system like overcrowded cities and decaying countryside. All these situations can be reproduced by modelling them using Cellular Automata (CA) [1, 2, 5]. CA models implement algorithms with simple rules and parameter controls, but the result can be a complex behaviour. A stability of naturally formed self‐organized urban system depends on its critical state parameter τ in the power law log(f(x)) = ‐τlog(x). If the system reaches self‐organized critical (SOC) state then it remains in it for a long time. The CA model URBACAM (URBAnistic Cellular Automata Model) describes the long‐lasting term behaviour and shows that the change in behaviour is sensitive to the urban parameter τ of the power law. Regionines urbanistines sistemos vystymasis išlieka viena iš opiausiu problemu žmonijos istorijoje. Keletas tokiu uždaviniu kaip miestu perpildymas, nykstančios kaimo vietoves ir t.t. gali būti nesunkiai modeliuojami naudojant lasteliu automatus (LA). LA metodas ypatingas tuo, kad realizuoja algoritma paprastu taisykliu bei parametru valdymo pagalba, tačiau rezultate galima gauti sudetinga elgsena. Natūraliai susiformavusiu urbanistiniu sistemu stabilumas priklauso nuo sistemos krizines savirangos būsenos (KSB) parametro τ. Jei sistema pasiekia KSB, tai ji ilga laika išlieka joje. LA modelis URBACAM charakterizuoja ilgalaike elgsena ir parodo, jog modelyje jos kitimus itakoja eksponentinio desnio urbanistinis parametras τ.

Bifurcation Dynamics and Pattern Formation of a Three-Species Food Chain System with Discrete Spatiotemporal Variables

2019 ◽  
Vol 74 (11) ◽  
pp. 945-959
Author(s):  
Huayong Zhang ◽  
Ge Pan ◽  
Tousheng Huang ◽  
Tianxiang Meng ◽  
Jieru Wang ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Food Chain ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Chain System ◽  
Self Organized ◽  
New Perspective ◽  
Centre Manifold Theorem

AbstractThe bifurcation dynamics and pattern formation of a discrete-time three-species food chain system with Beddington–DeAngelis functional response are investigated. Via applying the centre manifold theorem and bifurcation theorems, the occurrence conditions for flip bifurcation and Neimark–Sacker bifurcation as well as Turing instability are determined. Numerical simulations verify the theoretical results and reveal many interesting dynamic behaviours. The flip bifurcation and the Neimark–Sacker bifurcation both induce routes to chaos, on which we find period-doubling cascades, invariant curves, chaotic attractors, sub–Neimark–Sacker bifurcation, sub–flip bifurcation, chaotic interior crisis, sub–period-doubling cascade, periodic windows, sub–periodic windows, and various periodic behaviours. Moreover, the food chain system exhibits various self-organized patterns, including regular and irregular patterns of stripes, labyrinth, and spiral waves, suggesting the populations can coexist in space as many spatiotemporal structures. These analysis and results provide a new perspective into the complex dynamics of discrete food chain systems.

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