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A Biological Coevolution Model with Correlated Individual-Based Dynamics

Volkan Sevim? and Per Arne Rikvold?

School of Computational Science and Information Technology, Center for Materials Research and Technology, and Department of Physics, Florida State University, Tallahassee, FL 32306-4120, USA (Dated: February 9, 2008)

arXiv:q-bio/0403042v2 [q-bio.PE] 11 May 2004

We study the e?ects of interspeci?c correlations in a biological coevolution model in which organisms are represented by genomes of bitstrings. We present preliminary results for this model, indicating that these correlations do not signi?cantly a?ect the statistical behavior of the system.

I.

INTRODUCTION

The dynamics of biological coevolution poses many problems of interest to the statistical-physics and complexsystems communities [1, 2]. Recently, we studied a coevolution model that is a simpli?ed version of the one introduced by Hall, et al. [3, 4, 5, 6]. Our model, in which individuals give birth, mutate and die, displays punctuated equilibrialike, quiet periods interrupted by bursts of mass extinctions [7, 8]. Interactions in this model were given by a random interaction matrix. Here we report on a modi?ed version of the model, in which we have added correlations to the interaction matrix in order to increase the biological realism. The modi?ed model is compared with the original one to assess the e?ects of correlations.

II. MODEL

We used a bitstring genome of length L to model organisms in the Monte Carlo (MC) simulations [9, 10]. In this sense, each genotype, which is just an L-bit number, is considered a di?erent haploid species. Therefore, the terms “genotype” and “species” are used in the same sense in this paper. We denote the population of species I at a discrete time t as nI (t), and several of the 2L possible species can be present at the same time in our “ecosystem.” All species reproduce asexually (by cloning) in discrete, non-overlapping generations. In each generation t, every individual of species I is allowed to give birth to F o?spring with a probability PI . Whether it reproduces or not, the individual dies at the end of the generation, so that only o?spring can survive to the next generation. The reproduction probability for an individual of species I is given by [3, 4, 5, 6] 1 . M n ( t IJ J )/Ntot (t) + Ntot (t)/N0 ] J

PI ({nJ (t)}) =

1 + exp [?

(1)

The Verhulst factor N0 represents the carrying capacity of the “ecosystem” and prevents the total population Ntot (t) = I nI (t) from diverging to in?nity [11]. M is the interaction matrix, in which a matrix element MIJ represents the e?ect of the population density of species J on species I . A positive MIJ means that species I bene?ts from species J , while a negative MIJ corresponds to a situation in which species I is harmed or inhibited by the presence of J . The form of the interaction matrix M is discussed in the next section. In each generation, all individuals undergo mutation with a probability ?. If an individual is chosen to mutate, one bit in its genome is picked randomly and ?ipped. Since we consider di?erent genotypes as di?erent species, mutations lead to speciation, i.e., creation of another species.

? Electronic ? Electronic

address: sevim@csit.fsu.edu address: rikvold@csit.fsu.edu

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Single Correlation (Theoretical) Double Correlation (Theoretical) Single Correlation (Numerical) Double Correlation (Numerical)

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Uniform-uncorr. Gaussian-uncorr. Single Corr. Double Corr.

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FIG. 1: Correlation functions and matrix-element distributions after averaging. (a) Theoretical (solid) and numerical (dashed) results for the correlation functions for 213 × 213 interaction matrices of the single-correlated (lower) and double-correlated (upper) models. The theoretical and numerical results agree. (b) Distributions of matrix elements: uniform and Gaussian uncorrelated (solid curves with square and Gaussian distributions, respectively), single-correlated (dashed curve) and doublecorrelated (dot-dashed curve) models with L = 8. The approximately Gaussian distributions for the two correlated models and the distribution for the Gaussian-uncorrelated model practically overlap

III.

THE INTERACTION MATRIX

In this study, the interaction matrix M was set up in four di?erent ways. In the ?rst case, all o?-diagonal elements were randomly and uniformly distributed on [?1, 1], while all diagonal elements were set to zero. This corresponds to the case in Refs. [5, 6]. We shall call this the uniform-uncorrelated model. In the second case, we added correlations between interaction constants of similar species to make this model more realistic. In a real ecosystem, two di?erent but closely related species X and Y interact with another species Z in a similar way. Therefore, the interaction constant MXZ should be positively correlated with MY Z . To implement these correlations, we modi?ed the interaction matrix by averaging all terms over their nearest neighbors (nn) in Hamming space. Thus, ? ?

0 MIJ = ?MIJ +

(K,L)∈nn(I,J )

0 ? MKL /(2L + 1)1/2 ,

(2)

0 where MIJ are independent variables uniformly distributed on [?1, 1], and nn(I, J ) are those bitstring pairs that di?er from the bitstring pair (I, J ) by one bit (a Hamming distance of one). A square-root appears in the denominator because we multiply the average with the square-root of the normalization constant in order not to change the standard deviation of the matrix elements. To investigate the results of longer-range correlations we also modi?ed the random interaction matrix by averaging all terms over their nearest and next-nearest neighbors (nnn) in Hamming space:

0 MIJ = ?MIJ +

?

0 MKL + (K,L)∈nn(I,J ) (K,L)∈nnn(I,J )

0 ? /(2L2 + L + 1)1/2 MKL

?

(3)

where nnn(I, J ) is the set of bitstring pairs that di?er from (I, J ) by two bits. We shall call the nn-averaged and nnn-averaged models single-correlated and double-correlated, respectively. As a result of the central limit theorem, after averaging, the distributions of the matrix elements in both the single and double-correlated models take an approximately Gaussian form with the same standard deviation as the uniform distribution of the elements of the initial, random matrix (see Fig. 1b). In order to see whether a possible di?erence in the results for di?erent models is due to the correlations or the matrix-element distributions, we also set up another uncorrelated interaction matrix. In this fourth model, the uncorrelated, random matrix elements are distributed with 2

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Uniform-uncorrelated Gaussian-uncorrelated Single Correlation Double Correlation 2 1/x guide to the eye

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FIG. 2: Normalized histograms of species-lifetimes based on simulations of 225 generations each: uniform-uncorrelated (solid curve), Gaussian-uncorrelated (short-dashed curve), single-correlated (dot-dashed curve) and double correlated (long-dashed curve). The distributions for the models with a Gaussian matrix-element distribution overlap (within the margin of error), and they di?er signi?cantly from the lifetime distribution for the uniform-uncorrelated model. The histograms exhibit a power-law like decay with an exponent near ?2. Results are averaged over eight runs each

a Gaussian distribution with the same standard deviation, 1/3, as in the correlated models. We shall call this the Gaussian-uncorrelated model. The correlation functions and the distributions of interaction-matrix elements for all the models are shown in Fig. 1. The steps in the correlation functions are a result of the chosen metric. We use a city-block metric in which the Hamming distance between two matrix elements MIJ and MKL is given by H (I, K ) + H (J, L) where H (I, K ) is the Hamming distance between two L-bit bitstrings, I and K .

IV. SIMULATION RESULTS

The interaction matrix is set up at the beginning and is not modi?ed during the course of the simulation. For all models, we performed eight sets of simulations for 225 =33 554 432 generations with the same parameters as in Refs. [5, 6]: genome length L = 13, mutation rate ? = 10?3 per individual per generation, carrying capacity N0 = 2000, and fecundity F = 4. We began the simulations with 200 individuals of genotype 0. In order to compare the models, we constructed histograms corresponding to the species-lifetime distributions. The lifetime of a species is de?ned as the number of generations between its creation and extinction. As seen in Fig.2, the lifetime distributions of the correlated and Gaussian-uncorrelated models overlap within the margin of error, and they di?er signi?cantly from the lifetime distribution of the uniform-uncorrelated model. They all exhibit a power-law like decay with an exponent near ?2. The correlations between the matrix elements do not seem to a?ect the behavior of the lifetime distribution to a statistically signi?cant degree, at least not for the relatively weak correlations that were introduced here. On the other hand, changes in the marginal probability density of the individual matrix elements do have statistically signi?cant e?ects, even though gross features, such as the approximate 1/x2 behavior of the species-lifetime distribution are not changed. Similar conclusions are reached also for other quantities that we studied. In particular, the power-spectral density of the Shannon-Wiener species diversity index shows 1/f noise [5, 6] with an overall intensity that depends 3

more on the marginal matrix-element distribution than on correlations in M. Although we have tested only weak correlations, and so our conclusion is only preliminary, there appears to be no disadvantage in using a random interaction matrix to model such an “ecosystem.” This has obvious computational advantages as it makes it possible to simulate systems with larger numbers of completely di?erent species.

Acknowledgments

This research was supported by U.S. National Science Foundation Grant No. DMR-0240078, and by Florida State University through the School of Computational Science and Information Technology and the Center for Materials Research and Technology.

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