The Bak-Sneppen Model


This Java applet (If you don't see anything above, please install Java 1.4) models Darwin's equations on a 1 dimensional ecology of 100 interacting species S. Every species is assigned a fitness value F between 0 and 1. At each time instant the weakest species is killed together with its  #Neighbors (=2 nearest in standard version of model). They are replaced by new species with new random fitnesses. The ecology co-evolves to a highly correlated state, where fractal activity gives burst of originations and extinction, as seen in the fossil record of species on earth.

By activating mutations, the mutation rate can be adjusted between 0.001 and 1. A finite mutation rate will illustrate the different time-scales associated to evolving species with high or low F values.


Ammonite family three from 380million year BC to 64 million years ago (where they all went extinct).
From: N. Eldredge (1987).  Life Pulse, Episodes in the History of Life. New York. 246 pp. Pelican edition (Great Britain)


The evolution of species during the last 540 million years present some intriguing evidence for cooperative behavior: Often during the history of life there has been major "revolutions" where many species has been replaced simultaneously. Spectacular examples are the Cambrian explosion 540 million years ago where a huge variety of life arose in a short time interval, and the Cretaceous-Tertiary boundary where mammals took over large parts of the ecosystem.

Inbetween these major revolutions there have been periods of quiescence, where often all species seemed to live in "the best of all worlds", and only few species suffered extinctions. However, the pattern of life is more subtle than completely on-of transitions of macro-evolutionary revolutions. When inspecting the extinction record one observes that often there was extinction of smaller size in the more quiet periods. In fact if one plots the size distribution of ecological events, one observes all sizes of extinctions. That is, the large events is becoming gradually less frequent than smaller events. There is no "bump" or enhanced frequency for the large scale extinctions. However, the distribution is broader than a Gaussian, in fact close to scale free distribution 1/E^1.5. This overall gradual decline of event size distributions indicates:

1) That large and small events may be associated to similar type of underlying dynamics. If extinctions were external due to for example asteroid impacts (Alvarez et al (1987)), one would expect a peak at large events.

2) That the probability distribution for extinction events are non-Gaussian implies that the probability for obtaining large events are relatively large. This shows that the species in the ecosystem do not suffer extinction independently of each other. Thus the overall macroevolutionary pattern supports cooperativity, even on the scale of the global ecosystem.

In the figure we show the family tree of ammonites from their origination 350 million years ago until their extinction 66 million years ago. One observes a tree like structure, with branchings and killings, that occasionally undergoes a near total extinction. However, as long as a single species survive, one see that it can diversify and subsequent regain a large family. Finally 66 million years ago, not a single ammonite survived, and we now only know them from their very common fossils. What we cannot see on the family tree are the interactions between the species, nor can we see which other species these ammonite interacted with. Thus the dynamics behind their apparent nearly coherent

The toy model:

To model the observed macro evolutionary pattern we will start with objects on the size of the main players on this scale, lets call them species. A species of course consists of many individual organisms, and dynamics of species represent the coarse grained view of the dynamics of these entire populations. Thus, whereas population dynamics may be governed by some sort fitness, we propose that species dynamics is governed by stabilities.

Given the basic evolutionary entity, called a species, we characterize this with one number Fi. This number is the stability of the species on An ecosystem of species consists of selecting N numbers Fi each representing a species. Each of these species are connected to a few other species.

For simplicity, we put the numbers Fi,  i=1,2,...N on a 1-d line, corresponding to a 1-dimension ecosystem (the real ecosystem
will have a very high dimension, but model on such a geometry behave similarly). At each time-step we change the least stable species. In addition fitness/stability is defined in relative terms, which implies that the fitness of a given species is a function of the species it interacts with. The co-evolutionary updating rule then reads (Bak \& Sneppen (1994,1995)):

At each step, the smallest of the Fi are located. For this as well as its nearest neighbors one replaces their F's by new random numbers in the interval [0,1].

 It is the simplest model which exhibits a phenomenon called self organized criticality. As the system evolves, the smallest of the Fi's is eliminated. After a transient period, for a finite system, a statistically stationary distribution of F's is obtained. For very large N this distribution is a step function where the selected minimal F is always below or at a threshold value Fc and therefore the distribution of F constant above Fc. For  the dimension d=1 shown in the applet one typically we update two nearest neighbors, and then obtain a self-organized threshold Fc=0.6670. The upper panel in the applet panel illustrate how the minimal F sites moves in ``species space" as the system evolve. One observes a highly correlated activity, signaling that the system spontaneously developed some sort of cooperativity.
The selection of least stable species to mutate next implicitly assumes a separation of time scales in the dynamics. Thus the selection of the least fit of all species to change next is the natural outcome of the updating model:

At each step: Select each of the Fi with probability proportional to  exp(-Fi/r). This selection defines a list of active sites. For members in this list, replace them as well as their nearest neighbors by new random numbers in the interval [0,1].

Here r represent an attempt rate for microscopic evolutionary changes, and is proportional to the mutation rate per generation.  For low enough r this model degenerates into the one where always the minimal Fi is selected. The value of Fmin represents a time scale (in fact it is log( time)). Thus if Fmin is large, then exponentially long times pass without any activity at all.
If Fmin is small it is selected practically instantly.

Mutation rates r are typically chosen to be between 0.001 and 0.1.

Speciation and extinction events far beyond species boundaries is unknown. All we can say is that the behavior is far from what we would expect from a random asynchronous extinction/origination tree. some rather long evolutionary time scale (a time scale much longer that the organism reproduction time scale).


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