×

More rigorous results on the Kauffman-Levin model of evolution. (English) Zbl 1044.92042

Summary: The purpose of this note is to provide proofs for some facts about the NK model of evolution proposed by S. Kauffman and S. Levin [Towards a general theory of adaptive walks on rugged landscapes. J. Theor. Biol. 128, 11–45 (1987)]. In the case of normally distributed fitness summands, some of these facts have been previously conjectured and heuristics were given. In particular, we provide rigorous asymptotic estimates for the number of local fitness maxima in the case when K is unbounded. We also examine the role of the individual fitness distribution and find the model to be quite robust with respect to this.

MSC:

92D15 Problems related to evolution
60G50 Sums of independent random variables; random walks
60G35 Signal detection and filtering (aspects of stochastic processes)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Coddington, E. and Levinson, N. (1955). Theory of Ordinary Differential Equations . McGraw-Hill, New York. · Zbl 0064.33002
[2] Durrett, R. and Limic, V. (2003). Rigorous results for the NK model. Ann. Probab. 31 1713–1753. · Zbl 1049.60037
[3] Evans, S. and Steinsaltz, D. (2002). Estimating some features of NK fitness landscapes. Ann. Appl. Probab. 12 1299–1321. · Zbl 1040.60043
[4] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003
[5] Hirsch, M. and Smale, S. (1974). Differential Equations , Dynamical Systems , and Linear Algebra . Academic Press, New York. · Zbl 0309.34001
[6] Kauffman, S. (1993). The Origins of Order. Oxford Univ. Press.
[7] Kauffman, S. and Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128 11–45.
[8] Knight, F. (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc. Providence, RI. · Zbl 0458.60002
[9] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion , 2nd ed. Springer, New York. · Zbl 0804.60001
[10] Stanley, R. P. (1986). Enumerative Combinatorics , I. Wadsworth and Brooks/Cole, Belmont, CA. · Zbl 0608.05001
[11] Weinberger, E. (1991). Local properties of Kauffman’s NK model: A tunably rugged energy landscape. Phys. Rev. A 44 6399–6413.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.