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The model can be further complicated by adding spatial structure and migration,
ageing, etc. [74].
Simpler, phenomenological models may be useful in exploring how coherent phe-
nomena like mass extinction may arise from the self-organization of ecosystems.
Most of these models consider species (or niches) xed, disregarding mutation. Pos-
sible, the simplest one is the Bak-Sneppen (BS) model [75]. In the BS model, species
are simply assigned a location on a graph, representing the ecological relation be-
tween two species (in the simplest case, the graph is a regular one-dimensional
lattice with nearest-neighbor links). A species is assigned a random number, inter-
preted as its relative abundance.
i
At each time step, the smallest species is removed,
and its ecological place is taken by another one, with a random consistence. With
this change, the size of species connected to the replaced one are also randomly
changed. The smallest species at the following time step may be one of those re-
cently changed, or an unrelated one. The rst case is interpreted as an evolutionary
avalanche (mass extinction).
This extremal dynamics (choice of the smallest species) is able to self-organize
the ecosystem so that the size distribution of species is not trivial (a critical value
appears so that no smaller species are present), and the distribution of avalanche
size follows a power-law, similar (but with a dierent exponent) to that observed
in paleontological data by Raup [76]. This model is able to illustrate how self-
organization and punctuated equilibrium can arise internally in a dynamical system
with long-range coupling (due to the extremal dynamics).
The Bak-Sneppen model can be though as an extremal simplication of a food
web, which in turn can be considered a simplication of the microscopic dynamics
(individual-based) of a modeled ecosystem.
15.6. Conclusions
We have shown some aspects of a theoretical approach to self-organization in evo-
lutionary population dynamics. The ideal goals would be that of showing how
macro-evolutionary patterns may arise from a simplied individual-based dynam-
ics. However, evolutionary systems tend to develop highly correlated structure, so
that is it dicult to operate the scale separation typical of simple physical system
(say, gases). One can develop simple models of many aspects, in order to test the
robustness of many hypotheses, but a comprehensive approach is still missing.
References
[1] R. DawkinsThe Selsh Gene (Oxford University Press, Oxford UK 1989), p. 352;
S. Blackmore, The Meme Machine (Oxford University Press, Oxford UK 1999).
[2] There are many topics about the evolution of languages. For instance: L.L. Cavalli-
i
In the original paper this number is termed \tness", but in an almost-stable ecosystem all
individuals should have the same tness.
