Mean field theory for a simple model of evolution

Flyvbjerg, Henrik; Sneppen, Kim; Bak, Per

doi:10.1103/physrevlett.71.4087

Cited by 219 publications

(211 citation statements)

References 16 publications

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“…In the annealed case, the factor is replaced with p k simply. The last two terms do the addition of new fitness values [4]. The stationary solution in the limit t → ∞ can be solved by using ρ k (x) = − ∂ ∂x Q k (x) and taking the integral over [x, 1] of the whole formula as the integral equation,…”

Section: Rate Equation Approachmentioning

confidence: 99%

“…Let us denote by f k the probability that the vertex with the smallest fitness value has degree k. ρ k (x) is the distribution function of fitness values at the vertices with degree k. Here we set up the master equation for the fitness distribution for the quenched updating, following [4]. First we define a quantity Q k (x), the accumulative distribution of the fitness values at vertices with degree k:…”

Section: Rate Equation Approachmentioning

confidence: 99%

“…As done in [4], the threshold x c is determined by the comparison of the first term with the second term of Eq. (4) in their absolute magnitudes.…”

Section: Rate Equation Approachmentioning

confidence: 99%

“…A mean field version of the BS model was introduced [4], in which updated are the minimum fitness value as well as the fitness values of other K − 1 sites selected at random in the system. Such a modified model enables one to solve the problem analytically.…”

Section: Introductionmentioning

confidence: 99%

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Extremal dynamics on complex networks: Analytic solutions

2005

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The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold xc to be 1/( k f + 1), where k f = k 2 / k (= k ) in the quenched (annealed) updating case, where k n is the n-th moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent γ < 3 (γ > 3) for the quenched case in the thermodynamic limit. The theoretical value xc fits well to the numerical simulation data in the annealed case only. Avalanche size, defined as the duration of successive mutations below the threshold, exhibits a critical behavior as its distribution follows a power law, Pa(s) ∼ s −3/2 .

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Section: Rate Equation Approachmentioning

confidence: 99%

Section: Rate Equation Approachmentioning

confidence: 99%

“…As done in [4], the threshold x c is determined by the comparison of the first term with the second term of Eq. (4) in their absolute magnitudes.…”

Section: Rate Equation Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extremal dynamics on complex networks: Analytic solutions

2005

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“…Especially for system with extremal dynamics, very few exact results are available so far [5]. Most of our insight is based on numerical simulations, scaling arguments [6], or mean field solutions [7], related in general to random neighbor versions of the models.…”

mentioning

confidence: 99%

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

Montevecchi

Stella

2000

Phys. Rev. E

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In a semi-infinite geometry, a 1D, M -component model of biological evolution realizes microscopically an inhomogeneous branching process for M → ∞. This implies in particular a size distribution exponent τ ′ = 7/4 for avalanches starting at a free end of the evolutionary chain. A bulk-like behavior with τ ′ = 3/2 is restored if "conservative" boundary conditions strictly fix to its critical, bulk value the average number of species directly involved in an evolutionary avalanche by the mutating species located at the chain end. A two-site correlation function exponent τR ′ = 4 is also calculated exactly in the "dissipative" case, when one of the points is at the border. These results, together with accurate numerical determinations of the time recurrence exponent τ ′ f irst , show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as in the bulk, and their distribution exponents obey the basic scaling laws holding there.

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A Model for Evolution and Extinction

Roberts

Newman

1996

Journal of Theoretical Biology

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This article gives a brief introduction to the mathematical modeling of large-scale biological evolution and extinction. We give three examples of simple models in this field: the coevolutionary avalanche model of Bak and Sneppen, the environmental stress model of Newman, and the increasing fitness model of Sibani, Schmidt, and Alstrøm. We describe the features of real evolution which these models are intended to explain and compare the results of simulations against data drawn from the fossil record.

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Mean field theory for a simple model of evolution

Cited by 219 publications

References 16 publications

Extremal dynamics on complex networks: Analytic solutions

Extremal dynamics on complex networks: Analytic solutions

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

A Model for Evolution and Extinction

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