Diffusion and reaction in random media and models of evolution processes

Ébeling, W.; Engel, A.; Esser, B.; Feistel, Rainer

doi:10.1007/bf01011839

Cited by 83 publications

(30 citation statements)

References 10 publications

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“…branching. Other applications are: Fisher-Eigen equation in Darwinian evolution (Ebeling et al (1984)) Burgers' equation with a random force in hydrodynamics (Carmona and Molchanov (1994)). …”

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confidence: 99%

Correlation structure of intermittency in the parabolic Anderson model

Gärtner

Hollander

1999

Probab Theory Relat Fields

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Consider the Cauchy problem @u(x t)=@t = Hu(x t) ( x 2 Z Z d t 0) with initial condition u(x 0) 1 and with H the Anderson Hamiltonian H = + . Here is the discrete Laplacian, 2 (0 1) is a di usion constant, and = f (x): x 2 Z Z d g is an i.i.d. random eld taking values in IR. G artner and Molchanov (1990) have shown that if the law o f (0) is nondegenerate, then the solution u is asymptotically intermittent. This means that lim t!1 hu 2 (0 t )i=hu(0 t )i 2 = 1, where h i denotes expectation w.r.t. , and similarly for the higher moments. Qualitatively their result says that, as t increases, the random eld fu(x t): x 2 ZZ d g develops sparsely distributed high peaks, which g i v e the dominant c o n tribution to the moments as they become sparser and higher.In the present paper we study the structure of the intermittent peaks for the special case where the law o f (0) is (in the vicinity of) the double exponential Prob( (0) > s ) = exp ;e s= ] ( s 2 IR). Here 2 (0 1) is a parameter that can be thought of as measuring the degree of disorder in the -eld. Our main result is that, for xed x y 2 Z Z d and t ! 1 , the correlation coe cient o f u(x t) and u(y t) c o n verges to kw k ;2 2 (Z Z) with minimal l 2 -norm). Qualitatively our result says that the high peaks of u have a shape that is a multiple of w relative t o t h e c e n ter of the peak. It will turn out that if the right tail of the law o f (0) is thicker (or thinner) than the double exponential, then the correlation coe cient o f u(x t) and u(y t) c o n verges to x y (resp. the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.1991 Mathematics Subject Classi cation: 60H25, 82C44 (primary), 60F10, 60J15, 60J55 (secondary).

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confidence: 99%

Correlation structure of intermittency in the parabolic Anderson model

Gärtner

Hollander

1999

Probab Theory Relat Fields

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“…Under conditions of strong selection and small mutation rate, such a population will rapidly climb a local fitness maximum, where it then resides for a long time, until a rare, large fluctuation allows it to cross the valley to a more favorable peak. At least in the limit of infinite population size [5], the mathematics of this process is closely related to physical problems such as noise-driven barrier crossing, tunneling [6] and variablerange hopping [7], and it is easy to show that the residence time at one peak can be vastly larger than the time required for the transition to the next [8]. In a rugged fitness landscape, the sequence of transitions forms an evolutionary trajectory, which probes the distribution of fitness peaks and the geometry of the landscape.…”

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confidence: 99%

Punctuated evolution for the quasispecies model

Krug

Karl

2003

Physica A: Statistical Mechanics and its Applications

118

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Biological evolution in a sequence space with random fitnesses is studied within Eigen's quasispecies model. A strong selection limit is employed, in which the population resides at a single sequence at all times. Evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps. The distribution of the total evolution time displays a universal power law tail with exponent -2. Simulations show that the evolutionary dynamics is very well represented by a simplified shell model, in which the subpopulations at local fitness maxima grow independently. The shell model allows for highly efficient simulations, and provides a simple geometric picture of the evolutionary trajectories.Biological evolution often displays a punctuated dynamical pattern, in the sense that quiescent periods of stasis alternate with bursts of rapid change. A variety of mechanisms for punctuation have been proposed, which operate on different levels of the tree of life. On the largest scales of macroevolution, coevolutionary avalanches may play a role, which have been associated with self-organized criticality [1]. On the level of populations, punctuation due to rare, beneficial mutations has been observed in evolution experiments with bacteria [2]. Similar behavior has been found in simulations of RNA evolution, where stasis corresponds to diffusion on a neutral network, and a punctuation event marks the transition to another network of higher fitness [3].Possibly the simplest interpretation of punctuated evolution is in terms of a homogeneous population, represented by a localized distribution in some phenotypic or genotypic space, which evolves in a static, multipeaked fitness landscape [4]. Under conditions of strong selection and small mutation rate, such a population will rapidly climb a local fitness maximum, where it then resides for a long time, until a rare, large fluctuation allows it to cross the valley to a more favorable peak. At least in the limit of infinite population size [5], the mathematics of this process is closely related to physical problems such as noise-driven barrier crossing, tunneling [6] and variablerange hopping [7], and it is easy to show that the residence time at one peak can be vastly larger than the time required for the transition to the next [8]. In a rugged fitness landscape, the sequence of transitions forms an evolutionary trajectory, which probes the distribution of fitness peaks and the geometry of the landscape.In this Letter we investigate the statistics of such evolutionary trajectories in the framework of Eigen's quasispecies model [9,10]. We consider a population of individuals, each characterized by a binary genomic sequence σ of length N , which reproduce asexually and mutate in discrete time t. The total number of sequences is S = 2 N . An individual with genotype σ leaves A(σ) offspring in the next generation, and point mutations occur with probability µ per site and generation. In a mean field approximation, wh...

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“…We conclude (i) by suggesting that quasispecies can be conceived as a percolation model and (ii) forcibly stating the concept that there is an interesting analogy in the modeling of evolution processes on the one hand and disordered states on the other, which may support further investigation in this "eld of biophysics (Ebeling et al, 1984;Zel'dovich, 1938). The properties of &&rugged'' landscapes with multiple peaks and valleys resulting from the assumption that "tnesses take any values between zero and one have also been studied (Kau!man & Levin, 1987).…”

Section: Discussionmentioning

confidence: 96%