Tane S. Ray

A Predator–Prey Model with Genetics: Transition to a Self-Organized Critical State

A two species predator–prey model based on the Lotka–Volterra equations is proposed, where the fitness of an individual animal depends upon the relative strength of its genes. Simulations of the model show that the system passes from the standard oscillatory solution of the Lotka–Volterra equations into a steady-state regime, which exhibits many of the characteristics of self-organized criticality, including a 1/f power spectrum.

Is the fossil record indicative of a critical system

We examine the distribution of the relative fractional fluctuation of the magnetisation of a 3D Ising ferromagnet at the critical temperature Tc by considering a histogram of frequency of the fractional magnetisation. The distribution is highly non-Gaussian and is reminiscent of the histogram of extinction intensity for fossil genera shown by Raup1 (1991) (see also Newman (1996)). The graph of relative fractional change ofmagnetisation vs MC time is compatible in appearance with the graph showing the variation of the number of extinction of families of marine animals with time (over hundreds of millions of years) given by Raup (1991), Sepkoski (1993) or Newman (1996). Our results show a power law dependence usually taken as evidence for the existence of a critical system.

Damage in the low-temperature phase of the ±J spin glass in two to six dimensions

Indentical clones are made of configurations of the ±J spin glass at temperatures near and below the phase transition, and their damage is monitored as a function of temperature and time. Focal points with multiple bifurcations are found occurring near the onset of the phase transition temperature.

SPECIATION FROM EVOLUTION

Evolution, based on the principles of mutation and selection, is a powerful basis for microscopic changes which can account for the evolution of a species and macroscopic speciation where there is splitting of a species into two distinct new species. We show that a single species evolves into distinct species after several generations in an unrestricted genome space.

Self-organization of aging in a population approaching the steady state

The nonequilibrium asymptotic dynamics of a model for aging in a population of individuals initially having a random distribution of survival rates is studied. The model drives itself toward a steady state, and the average age tends toward a well-defined value. An analytic derivation shows that the average age of the members of the population decays in a power law fashion with the leading term of ordert−1. Monte Carlo simulations agree with the analytic work, and show that thet−1 decay is universally observed even when somatic mutations are introduced into the population.

Self-organised criticality in a genetic model of species-species interaction

Genetics incorporated into a two trophic level species-species interaction model allows the populations of species to evolve. The system enters a regime where the extinction of species follows an erratic pattern in time and appears chaotic to the eye. Quantitative measurements suggest that the dynamics of the model exhibits self organised criticality. Fourier Transform analysis of the time series and autocorrelation function for the population data, as well as the distribution of lineage sizes and longevity of lineages, all show power law behavior. The lineages are the analogue of avalanches in other models of self organised criticality

QUANTITATIVE LINK BETWEEN BIOLOGICAL EVOLUTION AND STATISTICAL MECHANICS

A model of evolution called the modified Wright–Fisher model (MWF) is introduced. It is shown to exhibit a second order phase transition, and a quantitative mapping is established between the mean field Ising model and itself. An equation of state and scaling function are derived for the MWF from the steady state solution of the governing quasispecies equations. The critical exponents are identical to those of the mean-field Ising model. Simulation data for the MWF on a two-dimensional square lattice show good evidence for a critical point. The susceptibility exponent is estimated and is found, within the uncertainty of the simulation data, to be equal to that of the two-dimensional Ising model, suggesting that the two models are in the same universality class.

Originations And Extinctions: Analyses Of Fossil Data And Simulations

We analyse the fossil data of Benton1 with and without interpolation schemes. By Fourier transform analysis, we find a frequency dependence of the amplitude of 1/f for the various interpolation schemes used in the past. We illustrate that shuffling the interpolated data changes the spectra only slightly. On the other hand, an identical analysis performed on the raw (uninterpolated) fossil data gives a flat frequency spectrum. We conclude that the 1/f behavior is an artifact of the interpolation schemes. We next introduce a simulation of extinctions driven only by interactions between two trophic levels. Fourier transform analysis of the simulation data shows a frequency dependence of 1/f. When the data are grouped into a form resembling the fossil record the frequency dependence vanishes, giving a flat spectrum. Our simulation produces a frequency spectrum that agrees with the observed fossil record.

Anomalous Biennial Oscillations in a Fisher Equation with a Discretized Verhulst Term

The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.