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there is no chance to observe twice the same genotype in evolution. Similar pheno-
types may emerge (convergent evolution). Due to the autocatalytic (either you eat
or you are eaten) character of ecosystems, once a branch of a bifurcation is taken,
the other alternative is forever forbidden (this generally corresponds to extinctions).
It is dicult to carry out a completely theoretical approach. Most of people
interested in theoretical results want to apply their results to the interpretation of
real biological systems. Secondly, in some 3 billion years, evolution has developed
such an intricate variety of \case studies" of applied evolution that we still have
a lot of poorly understood examples to be studied. Finally, the diculty in per-
forming experiments, discourages the study of eects that could be compatible with
evolution but have never been realized on the earth.
The theoretical approach allows to recognize that evolutionary dynamics applies
not only to biological system. There are at least other two environments that sup-
port (or may support) an evolutionary dynamics. The rst one is the human mind,
and in this case the replicators are concepts, ideas and myths that can be tracked
in human culture. Indeed, ideas may propagate from individual to individual, can
mutate and are selected for their \infectivity" and their tness to the cultural cor-
pus. Many cultural traits like religions proved quite capable of propagating, in spite
of the load and the detrimental eects they have on their hosts. The term meme
(sounding like gene and resembling memory) has been coined for this replicator [1].
A particularly important aspect of cultural evolution concerns languages, and in
particular language competition [2]. We have not enough space to exanime these
aspects here.
The second example is given by the giant computer network, Internet. Computer
viruses and worms are examples of replicating objects that can mutate and are
selected by their ability of escaping antiviral software and infecting other computers.
With the increasing power of computers and their pervasivity, it is expected that
such \life" forms will be much more common. Computer life forms and memes may
cooperate, think for instance to hoaxes, e-mail viruses, etc. We shall concentrate
here on \standard", \biological" life.
Some parts of this paper were previously published in Ref. [3]. See also Ref. [4].
15.2. The Structure of an Evolving System
The three basic blocks of evolutionary dynamics are replication, mutation and se-
lection. Evolution works on a population of reproducing individuals, often called
replicators.
15.2.1. Modeling living individuals
In order to visualize the problem, let us consider a pool of bacteria, growing in a
liquid medium in a well-stirred container, so that we do not have to be concerned
with spatial structures. We have chosen bacteria because they are able to synthesize
