Autocatalytic reactions in the isothermal, continuous stirred tank reactor

Gray, Peter; Scott, Stephen K.

doi:10.1016/0009-2509(83)80132-8

Cited by 532 publications

(238 citation statements)

References 11 publications

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“…Note that we are interested in the small-scale properties of the CARD model and thus in UV divergences, hence the restrictions on m and n (non-negative values) in Eqs. (11)- (12). Note also that the divergence structure of Γ ru depends on m and n, implying that it can be controlled externally via the noise exponents y v and θ v .…”

Section: Uv Renormalization Of the Stochastic Card Modelmentioning

confidence: 99%

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Effects of spatial and temporal noise on a cubic-autocatalytic reaction-diffusion model

2017

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We characterize the influence that external noise, with both spatial and temporal correlations, has on the scale dependence of the reaction parameters of a cubic autocatalytic reaction diffusion (CARD) system. Interpreting the CARD model as a primitive reaction scheme for a living system, the results indicate that power-law correlations in environmental fluctuations can either decrease or increase the rates of nutrient decay and the rate of autocatalysis (replication) on small spatial and temporal scales.

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Section: Uv Renormalization Of the Stochastic Card Modelmentioning

confidence: 99%

“…Of particular interest is the cubic autocatalytic reaction-diffusion (CARD) model [8,9], based on a two species autocatalytic chemical reaction [10][11][12][13][14][15]. Numerical simulations of the deterministic [8] and stochastic [9] CARD model show the appearance of self-replicating domains that are analogous to simple cells.…”

Section: Introductionmentioning

confidence: 99%

Effects of spatial and temporal noise on a cubic-autocatalytic reaction-diffusion model

2017

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“…We consider the Gray-Scott (GS) reaction diffusion model [7], which is one of the simplest models of biochemical relevance leading to spatial and temporal patterns when diffusion is included. Numerical simulations of this system in the deterministic case have revealed a surprisingly large set of complex and irregular patterns as a function of the system parameters [8].…”

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confidence: 99%

Dynamic renormalization group and noise induced transitions in a reaction diffusion model

Zorzano

Hochberg

Morán

2004

Physica A: Statistical Mechanics and its Applications

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We investigate how additive weak noise (correlated as well as uncorrelated) modifies the parameters of the Gray-Scott (GS) reaction diffusion system by performing numerical simulations and applying a Renormalization Group (RG) analysis in the neighborhood of the spatial scale where biochemical reactions take place. One can obtain the same sequence of spatial-temporal patterns by means of two equivalent routes: (i) by increasing only the noise intensity and keeping all other model parameters fixed, or (ii) keeping the noise fixed, and adjusting certain model parameters to their running scale-dependent values as predicted by the RG. This explicit demonstration validates the dynamic RG transformation for finite scales in a two-dimensional stochastic model and provides further physical insight into the coarse-graining analysis proposed by this scheme. Through several study cases we explore the role of noise and its temporal correlation in self-organization and propose a way to drive the system into a new desired state in a controlled way.

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“…To be specific, we will consider excitable (L = 0) GrayScott model, which is described by the following equations [20] …”

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confidence: 99%

Theory of spike spiral waves in a reaction-diffusion system

Muratov

Осипов²

1999

Phys. Rev. E

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We discovered a new type of spiral wave solutions in reaction-diffusion systems -spike spiral wave, which significantly differs from spiral waves observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of these waves in Gray-Scott model. We derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters. The theory does not rely on the specific features of Gray-Scott model and thus is expected to be applicable to a broad range of reaction-diffusion systems. PACS Number(s): 03.40. Kf, 05.70.Ln, 82.20.Mj, 82.40.Bj Formation of rotating spiral waves (rotors) is one of the most vivid and ubiquitous phenomena of nonlinear physics [1][2][3][4][5][6][7][8][9][10][11]. These waves are observed in nonlinear optical media [12], chemical reactions of BelousovZhabotinsky type, catalytic reactions on crystal surfaces [5][6][7], and in a variety of biological systems: social amoebae Dictyostelium discoideum [13], Xenopus oocytes [14], chicken retina [15], and the heart of animals and man, where the formation of spiral waves is responsible for cardiac arrythmias and the life-threatening condition of ventricular fibrillation [1,11].A generic model used to describe spiral waves is a pair of reaction-diffusion equations of activator-inhibitor type [1-10]where θ is the activator, i.e., the variable with respect to which there is a positive feedback; η is the inhibitor, i.e., the variable with respect to which there is a negative feedback and which controls activator's growth; q and Q are certain nonlinear functions representing the activation and the inhibition processes; l and L are the characteristic length scales, and τ θ and τ η are the characteristic time scales of the activator and the inhibitor, respectively; and A is the bifurcation parameter. A considerable amount of studies was done on the excitable systems with FitzHugh-Nagumo-type kinetics (N-systems) (see, for example, [1-8] and references therein). These systems are described by Eqs. (1) and (2) with L = 0, and the nonlinearity in q such that the nullcline of Eq.(1) is N-or inverted N-shaped. Theory of the spiral waves in N-systems with α = τ θ /τ η ≪ 1 was recently developed by Karma [16,17].The existence of spiral waves in excitable N-systems is due to the ability of such systems to sustain traveling waves, the simplest of which is a solitary wave -traveling autosoliton (AS) [1][2][3][4][5][6][7][8][9][10]. In N-systems the equation q(θ, η, A) = 0 has three roots: θ i1 , θ i2 , and θ i3 , for fixed A and η = η i . For α ≪ 1 an AS consists of a front, which is a wave of switching from the stable homogeneous state θ = θ h and η = η h to the state with θ = θ max (θ h = θ i1 and θ max = θ i3 for η i = η h ) whose width is of order l, and a back of width of order l that follows the front some distance w ≫ l behind the front. Thus, in the AS the distribution of θ is a broad pulse, while the value of η slowly varies from η = η h to some value η = η m in the back of the pulse, a...

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Autocatalytic reactions in the isothermal, continuous stirred tank reactor

Cited by 532 publications

References 11 publications

Effects of spatial and temporal noise on a cubic-autocatalytic reaction-diffusion model

Effects of spatial and temporal noise on a cubic-autocatalytic reaction-diffusion model

Dynamic renormalization group and noise induced transitions in a reaction diffusion model

Theory of spike spiral waves in a reaction-diffusion system

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