Criticality in a dynamics ruled evolutionary model (original) (raw)

A self-organized critical model for evolution

1996

A simple mathematical model of biological macroevolution is presented. It describes an ecology of adapting, interacting species. Species evolve to maximize their individual fitness in their environment. The environment of any given species is affected by other evolving species; hence it is not constant in time. The ecology evolves to a "self-organized critical" state where periods of stasis alternate with avalanches of causally connected evolutionary changes. This characteristic intermittent behaviour of natural history, known as "punctuated equilibrium," thus finds a theoretical explanation as a selforganized critical phenomenon. In particular, large bursts of apparently simultaneous evolutionary activity require no external cause. They occur as the less frequent result of the very same dynamics that governs the more frequent small-scale evolutionary activity. Our results are compared with data from the fossil record collected by J. Sepkoski, Jr., and others.

Self-organized criticality in a model of biological evolution with long-range interactions

Physica A: Statistical Mechanics and its Applications, 2000

In this work we study the effects of introducing long range interactions in the Bak-Sneppen (BS) model of biological evolution. We analyze a recently proposed version of the BS model where the interactions decay as r −α ; in this way the first nearest neighbors model is recovered in the limit α → ∞, and the random neighbors version for α = 0. We study the space and time correlations and analize how the critical behavior of the system changes with the parameter α. We also study the sensibility to initial conditions of the model using the spreading of damage technique. We found that the system displays several distinct critical regimes as α is varied from α = 0 to α → ∞ In recent years an increasing numbers of systems that present Self Organized Criticality [1,2] have been widely investigated. The general approach of statistical physics, where simple models try to catch the essencial ingredientes responsable for a given complex behavior has turned out to be very powerful for the study of this kind of problems. In particular Bak and Sneppen have introduced a simple model which has shown to be able to reproduce evolutionary features such as punctuacted equilibrium . Altough this model does not intend to give an accurate description of darwinian evolution, it catches into a single and very simple scheme (it is based on very simple dynamical rules) several features that are expected to be present in evolutionary processes, that is, punctuated equilibrium [3], Self Organized Criticality (SOC) [3] and weak sensitivity to initial conditions (WSIC) [5,6] (i.e., chaotic behaviour where the trajectories depart with a power law of the time instead of exponentially). In this sense, one important question arises about the robustness of such properties against modifications (i.e., complexifying) of the simple dynamical rules on which the model is based. The original model, hereafter referred as the first nearest neighbors (FNN) version [3], includes only nearest neighbors interactions in a one dimensional chain. This model presents SOC and weak sensitivity to initial conditions . On the other hand, another version of the model with interactions between sites randomly chosen in the lattice (and therefore it can be regarded as a mean field version of the FNN), hereafter referred as the random negihbors (RN) version [7], does not present SOC . Moreover, it is not expected (and we shall show in this work that it is indeed the case) to present WSIC.

Adaptive dynamics: a framework to model evolution in the ecological theatre

Journal of Mathematical Biology, 2010

The astonishing diversity of life has evolved over many millions of years of natural selection. Yet natural selection, as often taught in introductory courses and textbooks on population genetics, seems to explain why diversity should not exist: the survival of the fittest is the loss of everything else. Simple models of natural selection predict the fixation of the best allele in every locus. 1 Yet the Earth (or any part of it) is not ruled by a single Darwinian monster.

Evolution as a self-organized critical phenomenon

Proceedings of the National Academy of Sciences, 1995

We present a simple mathematical model of biological macroevolution. The model describes an ecology of adapting, interacting species. The environment of any given species is affected by other evolving species; hence, it is not constant in time. The ecology as a whole evolves to a "selforganized critical" state where periods of stasis alternate with

Extinction and self-organized criticality in a model of large-scale evolution

Physical Review E, 1996

A simple model of large-scale biological evolution is presented. This model involves an N-species system where interactions take place through a given connectivity matrix, which can change with time. True extinctions, with removal of less-fit species, occur followed by episodes of diversification. An order parameter may be naturally defined in the model. Through the dynamical equations, the system moves towards the critical threshold, which triggers the extinctions. The frequency distribution N(s) of extinctions of size s follows a power law N(s)Ϸs Ϫ␣ with ␣Ϸ2.3, close to known palaeobiological evidence.

A model of natural selection that exhibits a dynamic phase transition

Journal of Statistical Physics, 1987

A simple, stochastic model is developed of an asexual biological population that is undergoing natural selection. It is then observed that the size of the population, like the temperature parameter in the simulated annealing algorithm, is a measure of the amount of randomness to be allowed in the system. Exploiting the formal analogy between the two processes, it is shown that the distribution of different types of organisms in the population model converges to a stationary distribution if the population is growing more slowly than O(ln t) ("annealing"), but can fail to converge at all if the population is growing faster than O(ln t) ("quenching"). The results may be related to the "historical accidents" that permeate biological structures.

The Impact of Regulation in a Model of Evolving , Fitness-Maximising Agents

2001

The process of evolution of biological species has recently been analysed in formal models of complex systems. Individual species interact with each other, sometimes in cooperative symbiosis and sometimes in direct competition. The fitness level of each species evolves over time, and species whose fitness falls below a critical level become extinct. Variants of surviving species enter the eco-system in the niches vacated by extinct species. The overall properties of complex systems such as these emerge from the interactions of the individual species. A key property is that of self-organised criticality. The system becomes tuned so that extinctions on any scale can occur at any point in time. The probability of observing an extinction of any given size falls away not just with the size but with the size raised to a power of itself. The mathematical expression which describes the relationship between the size of an extinction and its frequency is known as a power law. Models based on ...

A Simple Model for the Evolution of Evolution

Journal of Biological Systems, 1997

A simple model of macroevolution is proposed exhibiting both the property of punctuated equilibrium and the dynamics of potentialities for different species to evolve towards increasingly higher complexity. It is based on the phenomenon of fractal evolution which has been shown to constitute a fundamental property of nonlinear discretized systems with one memory- or random-based feedback loop. The latter involves a basic "cognitive" function of each species given by the power of distinction of states within some predefined resolution. The introduction of a realistic background noise limiting the range of the feedback operation yields a pattern signature in fitness space with a distribution of temporal boost/mutation distances according to a randomized devil's staircase function. Introducing a further level in the hierarchy of the system's rules, the possibility of an adaptive evolutionary change of the resolution itself is implemented, thereby providing a time-depe...

On the scope of applicability of the models of Darwinian dynamics

arXiv (Cornell University), 2023

In their well-known textbook (Vincent & Brown, 2005), Vincent and Brown suggested an attractive approach for studying evolutionary dynamics of populations that are heterogeneous with respect to some strategy that affects the fitness of individuals in the population. The authors developed a theory, whose goal was to expand the applicability of mathematical models of population dynamics by including dynamics of an evolving heritable phenotypic trait subject to natural selection. The authors studied both the case of evolution of individual traits and of mean traits in the population (or species) and the dynamics of total population size. The authors consider the developed approach as (more or less) universally applicable to models with any fitness function and any initial distribution of strategies, which is symmetric and has small variance. Here it was shown that the scope of the approach proposed by Vincent & Brown is unfortunately much more limited. I show that the approach gives exact results only if the population dynamics linearly depends on the trait; examples where the approach is incorrect are given.

Stability and Explanatory Significance of Some Simple Evolutionary Models

Philosophy of Science, 2000

The explanatory value of equilibrium depends on the underlying dynamics. First there are questions of dynamical stability of the equilibrium that are internal to the dynamical system in question. Is the equilibrium locally stable, so that states near to it stay near to it, or better, asymptotically stable, so that states near to it are carried to it by the dynamics? If not, we should not expect to see this equilibrium. But even if an equilibrium is asymptotically stable, that is no guarantee that the system will reach that equilibrium unless we know that the system's initial state is sufficiently close to the equilibrium. Global stability of an equilibrium, when we have it, gives the equilibrium a much more powerful explanatory role. An equilibrium is globally asymptotically stable if the dynamics carries every possible initial state in the interior of the state space to that equilibrium. If an equilibrium is globally stable, it can have explanatory value even when we are completely uncertain about the initial state of the system. Once questions of dynamical stability are answered with respect to the dynamical system in question, there is the further question of structural stability of that system itself.