Hyperscaling in the Domany-Kinzel cellular automaton

Dickman, Ronald; Tretyakov, A. Yu.

doi:10.1103/physreve.52.3218

Cited by 58 publications

(96 citation statements)

References 9 publications

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“…These growing domains are separated by wide regions of C states. The growth process is similar to those observed in systems with two equivalent absorbing states [16,17,18,19]. Finally the present system develops into one of the symmetric two-species absorbing states D 13 or D 24 where ρ 1 = ρ 3 = (1 − ρ 0 )/2 and ρ 2 = ρ 4 = 0, or ρ 2 = ρ 4 = (1 − ρ 0 )/2 and ρ 1 = ρ 3 = 0.…”

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confidence: 80%

Phase transition and selection in a four-species cyclic predator-prey model

Szabó

Sznaider

2004

Phys. Rev. E

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We study a four species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all the four species if the concentration of vacant sites is lower than a threshold value. Above the treshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) and extend their common territory. According to our Monte Carlo simulations the observed phase transition is equivalent to those found in spreading models with two equivalent absorbing states although the present model has continuous sets of absorbing states with different portions of the two neutral species. The selection mechanism yielding symmetric phases is related to the domain growth process whith wide boundaries where the four species coexist.PACS numbers: 05.50.+q, 87.23.Cc Multispecies ecological models with spatial extension exhibit a large variety of possible stationary states as well as phase transitions when tuning the model parameters. In the original Lotka-Volterra models [1,2] as well as in the generalized versions the spatial distribution of species is neglected (see [3,4] for reviews). Now we report a phenomenon underlying the role of spatial effects in the biological evolution.In the simplest spatial Lotka-Volterra models the individuals of competitive species are residing on the sites of a lattice and the system evolution is governed by invasions along the nearest neighbor links. In many cases the species form domains whith growing sizes and sooner or later only one species will survive. Significantly different behavior is found if the species dominate cyclically each other, i.e., the corresponding food web is characterized by a directed ring graph [5,6,7]. Frachebourg and Krapivsky [6] have shown that fixation occurs if the number of species N s exceeds a threshold value N f (d) depending on the spatial dimension d. In this case the species form a frozen domain structure [6]. Conversely , the moving invasion fronts maintain a self-organizing polydomain structure. These patterns are widely studied for N s = 3 [5,8,9] because it can provide a stability against some external invaders for the spatial models [10,11,12]. Sato et al. [13] have shown that, if only one of the invasion rates differs from unity for even N s , then only the species with odd (even) labels survive . Very recently, the species biodiversity were studied by similar models in bacterial [14], phytoplankton [15] systems.In the above lattice models each site is occupied by an individual of the competitive species. Now we will consider a diluted version of these models for N s = 4. Namely, the sites may be empty and the individuals are allowed to jump to these empty sites. These elementary events can result in the formation of "defensive a...

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confidence: 80%

Phase transition and selection in a four-species cyclic predator-prey model

Szabó

Sznaider

2004

Phys. Rev. E

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“…Fig. 18) and the form of the hyperscaling law suggests that our case is the parity conserving version of the Compact Directed Percolation [29,39,40].…”

Section: Discussionmentioning

confidence: 99%

Damage spreading for one-dimensional, nonequilibrium models with parity conserving phase transitions

Ódor

Menyhárd

1998

Phys. Rev. E

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The damage spreading (DS) transitions of two one-dimensional stochastic cellular automata suggested by Grassberger (A and B) and the kinetic Ising model of Menyhárd (NEKIM) have been investigated. These non-equilibrium models exhibit non-directed percolation universality class continuous phase transition to absorbing states, exhibit parity conservation (PC) law of kinks and have chaotic to non-chaotic DS phase transitions, too. The relation of the critical point and the damage spreading point has been explored with numerical simulations. For model B the two transition points are well separated and directed percolation universality was found both for spin damage and kink damages in spite of the conservation of damage variables modulo 2 in the latter case. For model A and NEKIM the two transition points coincide with drastic effects on the damage of spin and kink variables showing different time dependent behaviours. While the kink DS transition of these two models shows regular PC class universality, the spin damage of them exhibits a discontinuous phase transition with compact clusters and PC-like spreading exponents. In the latter case the static exponents determined by finite size scaling are consistent with that of the spins of the NEKIM model at the PC transition point. The generalised hyper-scaling law is satisfied. Detailed discussion is given concerning the dependence of DS on initial conditions especially for the A model case, where extremely long relaxation time was found.

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“…General feature of these models is that dynamics takes place only at the boundaries. The action, which describes this behavior was proposed in (Dickman and Tretyakov, 1995;Peliti, 1986) …”

Section: Voter Model (Vm) Classesmentioning

confidence: 99%

Universality classes in nonequilibrium lattice systems

2004

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This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.

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Hyperscaling in the Domany-Kinzel cellular automaton

Cited by 58 publications

References 9 publications

Phase transition and selection in a four-species cyclic predator-prey model

Phase transition and selection in a four-species cyclic predator-prey model

Damage spreading for one-dimensional, nonequilibrium models with parity conserving phase transitions

Universality classes in nonequilibrium lattice systems

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