Title: Evolution 17If we consider the life history of an individual, it will include a probability that the individual survives to sexual maturity, a probability that it will meet with each other member of the population (which will take account of isolated subpopulation), a probability that, having met, it will mate fruitfully with the other individuals (which is presumably zero for juveniles and individuals of the same sex), and a probability associated with each of the possible genetic permutations that its offspring might carry.
Thus instead of connecting one individual in one generation to another individual in the next generation, the generational transformation operator connects the individual parent to a probability distribution of possible offspring.
If we tried to represent this geometrically the history of the species would be represented by a bundle of trajectories through a multi-dimensional phase space. We would be faced with the philosophical questions of probable descendants versus real live descendants which would require some kind of quantum analogies.
But we really aren't interested in that--we are interested in the relationship of the observed present population to a hypothetical past population represented by some fossil remains. If we start from the present individuals and project backwards (which we can do, in principle because there is nothing to say that the operators can't be inverted) we'd get a fuzzy probability distribution of what the ancestors of any present species might have been. But that "might have been" is pretty fuzzy unless we explicitly apply our evolutionary constraints.
What we are interested in is not the individuals that might have been the ancestors of a present individual, but the species that might have been ancestral to the present species. This sharpens the result considerably, because although the offspring of a given individual may lie anywhere in the space reachable by operator {E}, a lot of that space will represent offspring that will not be fit to survive in the sense of having descendants that are fit to survive.
If we take the probabilistic genetic operator and average it over the entire initial population of potential parents the result will be an operator which relates generations of populations, i.e., given an initial population having a certain probability distribution over all the possible genetic variations, the operator will transform this into a probability distribution over all the possible genetic variations in characteristics of the offspring generation.
Because of this multiple averaging the transformation operator can appear to be a simple operator under certain circumstances. In a stable environment it may appear to be a unit operator which transforms a population into an essentially identical population. This does not mean that any particular individual's genetic material will be represented in the next generation. It does mean that there will be certain kinds of characteristics representing "fitness" or adaptation to a particular environment that will be equally likely to appear in future generations.
Environmental factors can, however, make for pronounced changes in the population distribution.
The extreme case is selective breeding, where the probability of a successful mating is artificially maintained at zero except for certain subpopulation considered to be desirable. By the end of many generations of such selective breeding a population distribution can be quite different from the original population, as can be seen by comparing wild and domesticated species. The disadvantage is that the selective breeding also reduces the distribution in the genetic vector space to a narrow compass, so the domesticated species can be susceptible to new environmental factors like mutated disease species.
But barring selective breeding, we can assume that operators exist that relate ancestor generations to descendant generations.
We have defined these operators in terms of algebraic operators but they can also be represented geometrically in terms of transformations in a vector space. In particular, the net effect of the vector transformation operators can be geometrically represented by the population distribution in the genetic vector space. This is the approach that was used implicitly in the earlier sections. Thus this description in terms of operators that transform vectors is essentially equivalent to the geometric representation used previously, and all the arguments made there apply to the algebraic representation.
The disadvantage to using the genetic description of the individual as contained in DNA is that the DNA message is not yet an easily observable phenomenon. The more usual characterization of a species is the taxonomic description. In principle, however, the same kind of mathematical representation can be used.
In principle one can list all the distinguishing characteristics that a taxonomist might use for distinguishing any one species from any other and assign a number "1" if the characteristic is present and "0" if the characteristic is absent. One can then add to that ordered set of numbers a finite set of additional characteristics that can be used to describe the relevant intra-species variation between individuals of any particular species. The resulting ordered list of binary numbers constitutes a binary vector which can be used to describe the individuals in any given population.
If the science of genetics is more-or-less correct there will be a reasonable connection between this taxonomic vector and the genetic vector describing the individual's DNA chains. We may not be able to define this relationship in detail, but unless we assume that such a relationship exists we have to rethink genetics from scratch. Because of the way we represent the DNA and the taxonomy of the individual by vectors, the relationship takes the form of a transformation matrix.
If we believe that there is any systematic relationship between the characteristics of parents and children we have to believe that such a transformation matrix exists and that, in principle, we can apply it to the genetic vectors.
It is, therefore, as valid to describe the population distribution in terms of observable (or taxonomic) characteristics as it is to make the description in terms of DNA. This validates the notion of vector space distributions of individual characteristics that was the basis of the argument in the discussion on earlier pages.