### Abstract

We study the asymptotic value as L → ∞ of the time for evolution τ, understood as the first time to reach a preferred word of length L using an alphabet with N letters. The word is updated at unit time intervals randomly, but configurations with letters matching with the preferred word are sticky, i.e., the probability to leave the configuration equals 0 ≤ γ ≤1, where γ may depend on the configuration. The model is introduced in Ref.^{[} ^{5} ^{]} in the case γ = 0, where it was shown that E[τ] ∼ Nln (L). We first give an alternative proof of the logarithmic scale, by evaluating the mode of τ. We then answer positively a question posed by H. Wilf on whether τ is exponential when γ ≠ 0. The natural scaling γ =O(L ^{-1}) gives rise to several finite order limits, including the interacting model when γ depends linearly on the number of matches with the preferred word. The scaling limit of the number of non-matching letters follows a Galton-Watson process with immigration. In a related model, the empirical measure converges to the solution of a discrete logistic equation with possible nonzero steady state. In conclusion, the length of τ is a question of scaling.

Original language | English (US) |
---|---|

Pages (from-to) | 328-340 |

Number of pages | 13 |

Journal | Stochastic Models |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Jul 3 2013 |

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### Keywords

- Discrete logistic equation
- Evolution
- Fixation time
- Galton-Watson scaling limit

### ASJC Scopus subject areas

- Modeling and Simulation
- Statistics and Probability
- Applied Mathematics

### Cite this

*Stochastic Models*,

*29*(3), 328-340. https://doi.org/10.1080/15326349.2013.808908

**Fixation time for an evolutionary model.** / Grigorescu, Ilie; Kang, Min.

Research output: Contribution to journal › Article

*Stochastic Models*, vol. 29, no. 3, pp. 328-340. https://doi.org/10.1080/15326349.2013.808908

}

TY - JOUR

T1 - Fixation time for an evolutionary model

AU - Grigorescu, Ilie

AU - Kang, Min

PY - 2013/7/3

Y1 - 2013/7/3

N2 - We study the asymptotic value as L → ∞ of the time for evolution τ, understood as the first time to reach a preferred word of length L using an alphabet with N letters. The word is updated at unit time intervals randomly, but configurations with letters matching with the preferred word are sticky, i.e., the probability to leave the configuration equals 0 ≤ γ ≤1, where γ may depend on the configuration. The model is introduced in Ref.[ 5 ] in the case γ = 0, where it was shown that E[τ] ∼ Nln (L). We first give an alternative proof of the logarithmic scale, by evaluating the mode of τ. We then answer positively a question posed by H. Wilf on whether τ is exponential when γ ≠ 0. The natural scaling γ =O(L -1) gives rise to several finite order limits, including the interacting model when γ depends linearly on the number of matches with the preferred word. The scaling limit of the number of non-matching letters follows a Galton-Watson process with immigration. In a related model, the empirical measure converges to the solution of a discrete logistic equation with possible nonzero steady state. In conclusion, the length of τ is a question of scaling.

AB - We study the asymptotic value as L → ∞ of the time for evolution τ, understood as the first time to reach a preferred word of length L using an alphabet with N letters. The word is updated at unit time intervals randomly, but configurations with letters matching with the preferred word are sticky, i.e., the probability to leave the configuration equals 0 ≤ γ ≤1, where γ may depend on the configuration. The model is introduced in Ref.[ 5 ] in the case γ = 0, where it was shown that E[τ] ∼ Nln (L). We first give an alternative proof of the logarithmic scale, by evaluating the mode of τ. We then answer positively a question posed by H. Wilf on whether τ is exponential when γ ≠ 0. The natural scaling γ =O(L -1) gives rise to several finite order limits, including the interacting model when γ depends linearly on the number of matches with the preferred word. The scaling limit of the number of non-matching letters follows a Galton-Watson process with immigration. In a related model, the empirical measure converges to the solution of a discrete logistic equation with possible nonzero steady state. In conclusion, the length of τ is a question of scaling.

KW - Discrete logistic equation

KW - Evolution

KW - Fixation time

KW - Galton-Watson scaling limit

UR - http://www.scopus.com/inward/record.url?scp=84882320892&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882320892&partnerID=8YFLogxK

U2 - 10.1080/15326349.2013.808908

DO - 10.1080/15326349.2013.808908

M3 - Article

AN - SCOPUS:84882320892

VL - 29

SP - 328

EP - 340

JO - Stochastic Models

JF - Stochastic Models

SN - 1532-6349

IS - 3

ER -