What is the working mathematical formula for evolution

Evolution is anything but random and there are several mathematical models that describe the process. The most fundamental is the Hardy-Weinberg Equilibrium equation. It is the primary mathematical null hypothesis for evolutionary theory and describes allele frequencies with the equation:

p² + 2pq + q² = 1

Where p is the frequency of allele "A" and q is the frequency of allele "a"

The model shows that evolution will not occur in a population if seven conditions are met:

1. mutation is not occurring
2. natural selection is not occurring
3. the population is infinitely large
4. all members of the population breed
5. all mating is totally random
6. everyone produces the same number of offspring
7. there is no migration in or out of the population

1) If mutation occurs, the allele frequencies will change.
2) If there is a selection event on the trait (e.g. drought, habitat destruction, etc) the allele frequencies will change.
3) If the population is infinitely large, the allele frequencies will equal infinity; clearly ALL populations of organisms are finite.
4) Breeding is competitive and rarely if ever do 100% of the members of a population breed and produce offspring. If some members are more successful than others at breeding, allele frequencies will change.
5) If mating is non-random, those individuals who have more offspring will increase their allele frequency in the population; traits that favor reproduction will out-pace those that do not and allele frequencies will change.
6) If various breeding pairs produce more or less offspring than others, allele frequencies will change.
7) If members of the species from other populations migrate into the population (or breeding members migrate out) the allele frequencies will change.