There are a number of aspects which characterise the selection, convergence and fitness in standard genetic algorithms [118, 215] . Chief among these are the loss of diversity, reproduction rate, and takeover. We will briefly characterise these in temporal terms.
The loss of diversity may be readily transcripted.
Loss of diversity is related to the variance of the fitness in the
population. It can be shown [368] that, for proportionate selection
(e.g. roulette wheel) the mean fitness 
of a population
changes depending on the variance 
as follows
In other words, mean fitness changes less and less as the genetic algorithm
converges, unless we can ensure that variance remains high. Note that
although equation 
only applies to roulette wheel
selection, although the broad statement ``improvement is dependent on
variance'' remains true [368] . Note that this applies to selection
only; since the genetic algorithm is a dynamic process, with three parts: an
exploratory process, a selective process, and a disruptive process the above
statement only holds when the effect from selection only is examined.
Mutation, i.e. the disruptive process, would tend to invalidate the above,
albeit with a typically very small probability.
In order to better characterise ratios of individual characteristics, especially regarding fitness, we need to be able to lump individuals with almost equal fitnesses together.
In general, a fitness function 
provides a value for each
individual used to grade them; this value is denoted 
(because it
solely depends on the chromosome in the individual as previously remarked).
The problem is that this value is often too fine-grained; in other words,
too discerning for applying a simple larger-than test. This is especially
true when we want to be able to compare individuals in the fitness space, as
opposed to phenotype space. If this is the case we may `coarsen' it by
defining a range within which two fitness values are considered ``equal
enough'' to be considered the same. Thus we define a equality environment as
follows.
Similarly, the set 
of individuals in an equality environment
is straightforwardly defined as
The reproduction rate may now be characterised for an equality environment. This definition essentially parallels the one in [118] .
From a temporal point of view the selection intensity above is not too meaningful. Instead, as the basic idea behind the selection intensity, namely the progress toward fitter and fitter individuals simultaneously implies that the existing individuals will be more and more difficult to replace. This, in turn, implies that the age will increase - and that the proportion of survived individuals from a previous interval will increase. We define the survival rate as follows.
Goldberg and Deb [215] define the takeover time as the time before the population
consists of (at least) 
copies of the best individuals, measured
using the fitness. In our context this implies that the
corresponding fitness space individuals should belong to the same equality
environment since they by definition do not have equal fitnesses.
Looking at the situation from a fitness perspective, takeover has occurred
when a sufficient number of individuals are in the same equality
environment. At that time, and in following intervals, the fitness profile
(i.e. the actual equality environments present in the population) will no
longer change, but stay constant. In other words, the genotypes of the
individuals may change but their fitnesses stay in the same equality
environment. What this means is that since the takeover time as defined by
[215] is dependent on the selection mechanism (and
is indeed different for each one) we cannot define the takeover without
specifying the selection used. However, a characterisation based solely on
fitness would not be dependent on the selection mechanism, provided the aim
is to maintain or improve the fitness of the population. So we say that
takeover has occurred when all individuals in the population belong to the
same equality environment. This is formalised in definition .
Unfortunately, this equation does not help us in formulating an estimate for the takeover time (à la Goldberg and Deb in [215] ). That result depends on the selection mechanism, and assumes a fitness function behaving in a certain way (either linearly or exponentially). No such assumptions are possible here.
