A physical model for competition between biological speciation and extinction

Introducing the effect of extinction into the so-called replicator equations in mathematical biology, we construct a general model of ecosystems. The present model shows mass extinction by its own extinction dynamics when the system initially has a large number of species ( diversity). The extinction dynamics shows several significant features such as a power law in basin size distribution, induction time, etc. The present theory can be a mathematical foundation of the specie...

Fluctuations in diversity and extinction sizes are discussed and compared for two different, individual-based models of biological coevolution. Both models display power-law distributions for various quantities of evolutionary interest, such as the lifetimes of individual species, the quiet periods between evolutionary upheavals larger than a given cutoff, and the sizes of extinction events. Time series of the diversity and measures of the size of extinctions give rise to fli...

We present an individual based model of evolutionary ecology. The reproduction rate of individuals characterized by their genome depends on the composition of the population in genotype space. Ecological features such as the taxonomy and the macro-evolutionary mode of the dynamics are emergent properties. The macro-dynamics exhibit intermittent two mode switching with a gradually decreasing extinction rate. The generated ecologies become gradually better adapted as well a...

We use a combination of analytic models and computer simulations to gain insight into the dynamics of evolution. Our results suggest that certain interesting phenomena should eventually emerge from the fossil record. For example, there should be a ``tortoise and hare effect'': Those genera with the smallest species death rate are likely to survive much longer than genera with large species birth and death rates. A complete characterization of the behavior of a branch of the p...

For taxonomic levels higher than species, the abundance distributions of number of subtaxa per taxon tend to approximate power laws, but often show strong deviationns from such a law. Previously, these deviations were attributed to finite-time effects in a continuous time branching process at the generic level. Instead, we describe here a simple discrete branching process which generates the observed distributions and find that the distribution's deviation from power-law form...

Twenty years ago, after analysing palaeontological data, Raup and Sepkoski suggested that mass extinctions on Earth appear cyclically in time with a period of approximately 26 million years (My). To explain the 26My period, a number of proposals were made involving, e.g., astronomical effects, increased volcanic activity, or the Earth's magnetic field reversal, none of which, however, has been confirmed. Here we study a spatially extended discrete model of an ecosystem and sh...

A class of models for large-scale evolution and mass extinctions is presented. These models incorporate environmental changes on all scales, from influences on a single species to global effects. This is a step towards a unified picture of mass extinctions, which enables one to study coevolutionary effects and external abiotic influences with the same means. The generic features of such models are studied in a simple version, in which all environmental changes are generated a...

This article gives a brief introduction to the mathematical modeling of large-scale biological evolution and extinction. We give three examples of simple models in this field: the coevolutionary avalanche model of Bak and Sneppen, the environmental stress model of Newman, and the increasing fitness model of Sibani, Schmidt, and Alstrom. We describe the features of real evolution which these models are intended to explain and compare the results of simulations against data dra...

Population dynamics in random ecological networks are investigated by analyzing a simple deterministic equation. It is found that a sequence of abrupt changes of populations punctuating quiescent states characterize the long time behavior. An asymptotic analysis is developed by introducing a log-scaled time, and it is shown that such a dynamical process behaves as non-steady weak chaos in which population disturbances grow algebraically in time. Also, some relevance of our st...

We consider a continuum version of a previously introduced and numerically studied model of macroevolution (PRL 75, 2055, (1995)) in which agents evolve by an optimization process in a rugged fitness landscape and die due to their competitive interactions. We first formulate dynamical equations for the fitness distribution and the survival probability. Secondly we analytically derive the $t^{-2}$ law which characterizes the life time distribution of biological genera. Thirdly...