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
Fixation time for an evolutionary model. / Grigorescu, Ilie; Kang, Min.
In: Stochastic Models, Vol. 29, No. 3, 03.07.2013, p. 328-340.Research output: Contribution to journal › Article
}
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 -