Pattern formation and periodic structures in systems modeled by reaction-diffusion equations

Greenberg, J. M.; Hassard, Brian; Hastings, S. P.

doi:10.1090/s0002-9904-1978-14560-1

Cited by 111 publications

(37 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An extremely simple cellular three-state model of an excitable medium with interactions between neighboring cells was studied by , and shown to exhibit spiral patterns. See also (Greenberg, Hassard, and Hastings (1978)). In a model with this simplicity, one is also able to gain insight into the reason for the appearance of spirals.…”

Section: Pulsesmentioning

confidence: 98%

Asymptotic states for equations of reaction and diffusion

Fife¹

1978

Bull. Amer. Math. Soc.

100

View full text Add to dashboard Cite

Section: Pulsesmentioning

confidence: 98%

Asymptotic states for equations of reaction and diffusion

Fife¹

1978

Bull. Amer. Math. Soc.

100

View full text Add to dashboard Cite

“…The CA rules indicate how a certain amount of liquid is transferred, and define the amount of liquid loss as a result of the infiltration process. The rules for the spreading of liquids are based on the use of the Manning coefficient, where the fluid velocity v in each cell in which the liquid is present is calculated according to the following equation (5).…”

Section: The Rules and Conditionsmentioning

confidence: 99%

“…The Manning roughness coefficient n accounts for the effect of bed roughness on the flow field, and its determination is essential to the accuracy of the calculated flow, sediment transport, and bed change. [12] The velocity is calculated using relation (5) and is used to determine the time (6) needed to move the liquid through the cell.…”

Section: The Rules and Conditionsmentioning

confidence: 99%

“…The original use of cellular automata was in biology as an idealization of the biological system [15,16]. Other applications where cellular automata have been used are in non-linear chemical systems [5], model design for the development of spiral galaxies [4,13], and in the modelling of intensification of dendritic crystals [7]. The cellular automaton developed for this paper will cover an application that affects the environment around us, namely the above-mentioned simulation of the water mass in the form of runoff water across the terrain.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cellular Automata for the Flow Simulations on the Earth Surface, Optimization Computation Process

Cirbus¹,

Podhoranyi²

2013

Appl. Math. Inf. Sci.

View full text Add to dashboard Cite

Abstract:The spreading of water across terrain is not a simple and isolated process. The spread of water is a part of the more general water cycle. Many commercial software exist for water modelling, but their correct use requires a considerable knowledge of water modelling, and about the study area. Using the relatively simple principle of cellular automata (CA) it is possible to obtain results which compare well with real measurements. This article describes using a CA for simulating the spreading of liquid, using comparatively simple rules and conditions which include several factors affecting the spreading of water such as slope, roughness and infiltration. A disadvantage of CA is the slower calculation process, which is strongly dependent on the size of the study area. We address this issue by using three optimization methods to reduce the computation time.

show abstract

“…The model we study is a stochastic variation of the Greenberg-Hastings cellular automaton [5] (GHCA). In spite of its apparent simplicity, this model displays complex behaviors.…”

Section: Introductionmentioning

confidence: 99%