Evolution in a nutshell

an altrnative outline on evoution

and some consequences concerning valuations

by

Gregor Kjellström

            References          

Six theorems in control of evolution; high school level.

In short, there is a pocketful of mathematical theorems ruling the evolution, at least at a fairly good statistical second order approximation. This means that the gene pool is approximated by a normal distribution. Then by using the rules of genetic variation ( crossing over, inversion etcetera) as a random number generator, evolution may effectuate a simultaneous maximization of mean fitness and phenotypic disorder/diversity. This means that evolution strives to secure our survival with the largest possible margins to spare while the disorder stands for imagination and creativity. 

1   The central limit theorem: The sum of a large number of random steps tend to become normally Gaussian distributed. See Cramér in references.

      Since the development from fertilized egg to adult individual may be seen as a stepwise modified repetition of the evolution of a particular individual, morphological characters (parameters) tend  to become normally distributed. As examples of such parameters we may mention the length of a bone or the distance between the pupils. Even mental parameters such as IQ may also be normally distributed.

2    The theorems of normal adaptation: If the centre of gravity (m*) of the gene pool of the parents to offspring in the next generation coincides with the centre (m) of the normally distributed gene pool in the next generation – in a state of selective equilibrium (m* = m) - then the mean fitness is maximal. See theorem 6.2.2.

      The theorem may be proved in two different ways. Firstly, one may maximize mean fitness while keeping the disorder of the normal distribution constant. Secondly, one may maximize the disorder of the normal distribution keeping the mean fitness constant. In both cases the condition of optimality will be the same, m* = m. This means that evolution effectuates a simultaneous maximization of mean fitness and phenotypic disorder/diversity.

A more general formulation of the theorem includes the mean value of information and the moment matrix M of the normal distribution allowing he disorder to increase even more, still keeping the mean fitness constant. The condition of optimality becomes M* proportional to M, where M* is the moment matrix for the parental distribution. This will make normal adaptation a second order approximation of evolution.

Note that mean fitness is calculated as a mean over the set of individuals, in contrast to the fundamental theorem of biology (Fisher, 1930) where mean fitness is calculated over the set of genes leading to a theorem where the duality between mean fitness and mean information is missing.

3    A theorem about disorder: The normal distribution is the most disordered distribution among all statistical distributions having the same variance. See theorem 6.2.4.

4    The theorem for the choice of a breeding partner (Hardy-Weinberg): If mating takes place at random, then the allele frequencies in the next generation are exactly the same as they were for the parents. See Hartl, 1981.

      Since the centers of gravity (m* and m) will behave similarly, evolution will strive to fulfill the condition of optimality of the theorem of normal adaptation according to point 2, and mean fitness will be maximized.

5    The second law of thermodynamics (the entropy law): The disorder will always increase in all isolated systems. But in order to avoid considering isolated systems I prefer an alternative formulation: A system attains its possible states in proportion to their probability of occurrence. See also references, Reif, 1985 and Brooks, 1986.

      Thus, the system will attain its most probable disordered states of occurrence even if some force from outside influences it. If the mutation rate is sufficiently high, evolution will also be able to maximize- at least to some extent - the phenotypic disorder/diversity in accordance with the theorem of normal adaptation, point 2, see Kjellström, 1996 in references.

6    The theorem of efficiency. The most important difference between the natural and the simulated evolution in my PC is that the natural one is able to test millions of individuals in parallel, while my PC has to test one at a time.

But the efficiency of evolution also depends on the mutation rate and this is plain from the theorem of efficiency. It is based on the theory of information (Shannon 1948, see Middleton). So, if P is the probability that an individual in a large population will be able to survive, then the negative logarithm of P, -log(P), is the information in the art of survival gained when a survivor has been found. Since the inverse of P is proportional to the work or time needed to find a survivor, then –P*log(P) becomes a measure of efficiency.

A simplified version of the theorem states that all measures of efficiency, that satisfy certain postulates, are asymptotically proportional to -P*log(P) when the number of statistically independent parameters tend towards infinity. See On the efficiency of random search and references Kjellström, 1991.

Maximum efficiency is attained when P = 1/e = 0.3679, where e is the base of the natural logarithmic system. As an example of a measure (not based on the theory of information) of efficiency I may mention the average speed of a random walk in a simplex region. The average speed will asymptotically tend to -P*log(P) when the number of dimensions tend towards infinity. See references Kjellström, 1969.