The inactive–active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

Mendonça, J. Ricardo G.

doi:10.1142/s0129183116500169

Cited by 6 publications

(3 citation statements)

References 41 publications

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“…In the one-site mean field approximation, all correlations are written in terms of one-site correlation. That is, the probability of any cluster of sites is written as the product of the probability of each site (Tomé and De Carvalho 2007;Mendonça 2016). Also, all correlations between sites in the cluster are neglected in one-site mean field approximation (Tomé and De Carvalho 2007).…”

Section: Mean Field Approximationmentioning

confidence: 99%

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

Ruhomally

Dauhoo

2020

Comp. Appl. Math.

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A stochastic process depicting the spreading dynamics of illicit drug consumption in a given population forms the crux of this work. A probabilistic cellular automaton (PCA) model is developed to examine the effects of the social interactions between nonusers and drug users. The model, called NERA, comprises four classes of individuals, namely, nonuser (N), experimental user (E), recreational user (R), and addict user (A). The stochastic process evolves in time by local transition rules. By means of dynamical simple mean field approximation, a nonlinear system of differential equations illustrating the dynamics of the PCA is obtained. The existence and uniqueness of a positive solution of the model is established, and the fixed points of the system are sought to perform the stability analysis. Furthermore, a stochastic mean field (SMF) approach to the NERA system is introduced. SMF extends the latter model to integrate the stochastic behaviour of drug consumers in a given environment. The SMF system is shown to exhibit a unique global solution which is stochastically ultimately bounded. Simulations of the cellular automaton and mean field analysis are used to study the evolution of the model. Verification and validation are carried out using data available on the consumption of cannabis in the state of Washington (Darnell and Bitney in I − 502 evaluation and benefit-cost analysis: second required report, Washington state institute for public policy. Technical Report, 2017). These numerical experiments confirm that the NERA model can help in the analysis and quantification of the spatial dynamics of illicit drug usage in a given society and eventually provide insight to policy-makers on different steps to be taken to curb this social epidemic.

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Section: Mean Field Approximationmentioning

confidence: 99%

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

Ruhomally

Dauhoo

2020

Comp. Appl. Math.

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“…The PCA in table 1 is an example of a mixed PCA, also known as 'diploid' cellular automata [28][29][30][31][32][33]. In a mixed PCA, two or more deterministic CA rules are combined such that sometimes one rule is applied, some other times another rule is applied.…”

Section: A Mixed Pca Inspired By Population Dynamicsmentioning

confidence: 99%

A probabilistic cellular automata model for the dynamics of a population driven by logistic growth and weak Allee effect

Mendonça¹

2018

J. Phys. A: Math. Theor.

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We propose and investigate a one-parameter probabilistic mixture of one-dimensional elementary cellular automata under the guise of a model for the dynamics of a single-species unstructured population with nonoverlapping generations in which individuals have smaller probability of reproducing and surviving in a crowded neighbourhood but also suffer from isolation and dispersal. Remarkably, the first-order mean field approximation to the dynamics of the model yields a cubic map containing terms representing both logistic and weak Allee effects. The model has a single absorbing state devoid of individuals, but depending on the reproduction and survival probabilities can achieve a stable population. We determine the critical probability separating these two phases and find that the phase transition between them is in the directed percolation universality class of critical behaviour.

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“…The one-parameter PCA in Case II are examples of mixed PCA that have appeared in the literature before [54,56,57]. In a mixed PCA, two or more deterministic CA rules are combined probabilistically such that sometimes one rule is applied, some other times another rule is applied to a given cell.…”

Section: B Mixed Ca and Pcamentioning

confidence: 99%

Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects

Mendonça

Gevorgyan

2017

Phys. Rev. E

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We investigate one-dimensional elementary probabilistic cellular automata (PCA) whose dynamics in first-order mean-field approximation yields discrete logisticlike growth models for a single-species unstructured population with nonoverlapping generations. Beginning with a general six-parameter model, we find constraints on the transition probabilities of the PCA that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof. Several possible models display a negative cubic term that can be interpreted as a weak Allee factor. We also investigate the conditions under which a one-parameter PCA derived from the more general six-parameter model can generate valid population growth dynamics. Numerical simulations illustrate the behavior of some of the PCA found.

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The inactive–active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

Cited by 6 publications

References 41 publications

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

A probabilistic cellular automata model for the dynamics of a population driven by logistic growth and weak Allee effect

Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects

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