Complex and Adaptive Dynamical Systems by Claudius Gros

Author:Claudius Gros
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(6.39)

We note that Q(x) ∈ [0, 1] and that Q(0) = 1, Q(1) = 0. There must therefore be some x ∈ ]0, 1[ for which 0 < Q(x) < 1. Then

(6.40)

Equation (6.40) remains valid as long as Q < 1, or x > x c :

We then have in the limit N → ∞

(6.41)

compare Fig. 6.15, and, using ,

(6.42)

This result compares qualitatively well with the numerical results presented in Fig. 6.14. Note, however, that the mean-field solution Eq. (6.42) does not predict the exact critical barrier height, which is somewhat larger for K = 2 and a one-dimensional arrangement of neighbors, as in Fig. 6.14.

Fig. 6.15The distribution Q(x) to find a fitness barrier larger than x ∈ [0, 1] for the Bak and Sneppen model, for the case of random barrier distribution (dashed line) and the stationary distribution (dashed-dotted line), compare Eq. (6.41)

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