User:HowieKor/sandbox

Test of text under picture

Models

The Evolutionary Game Model

Evolutionary Game Theory transposes Darwinian mechanisms into a mathematical form by adopting a System Model of evolutionary processes with three main components - Population, Game, and Replicator Dynamics. The system process itself has four phases:

1) The model (as evolution itself) deals with a Population (Pn). The population will exhibit Variation among Competing individuals. In the model this competition is represented by the Game.

2) The Game tests the strategies of the individuals under the “rules of the game”. These rules produce different payoffs – in units of Fitness (the production rate of offspring). The contesting individuals meet in pairwise contests with others, normally in a highly mixed distribution of the population. The mix of strategies in the population affects the payoff results by altering the odds that any individual may meet up in contests with various strategies. The individuals leave the game pairwise contest with a resulting fitness determined by the contest outcome – generally represented in a Payoff Matrix.

3) Based on this resulting fitness each member of the population then undergoes replication or culling determined by the exact mathematics of the Replicator Dynamics Process. This overall process then produces a New Generation P(n+1). Each surviving individual now has a new fitness level determined by the game result.

4) The new generation then takes the place of the previous one and the cycle begins again (and never stops). Mathematically speaking it is an Iterative process. Over time the population mix in such a system may converge to a stationary state – and if such a state cannot be invaded by any new “mutant strategies” it is by definition and Evolutionary Stable State (ESS)

]]

It is important to realise that EGT is not just a specialist mathematical treatment of animal contests to determine their dynamics and results, but in a manner similar to the field of Evolutionary Algorithms, it is a comprehensive model that encompasses the Darwinian process itself – including the central tenets of competition (the game) natural selection (replicator dynamics) and heredity - all within the overall model. Therefore it is a major vehicle to help understand and explain some of the most fundamental questions in biology including the issue of group selection, sexual selection, altruism, parental care, co evolution, and ecological dynamics. Much of the progress in developing understanding in these diverse areas has been aided by Evolutionary Game Theory modelling and many of the counter intuitive situations in these areas have been explained and put on a firm mathematical footing by the use of these models.

Taking the PAYOFF MATRIX results and plugging them into the above equation:

WHawk= V*(1-p)+(V/2-C/2)*p

Similarly for a Dove:

WDove= V/2*(1-p)+0*(1-p)

so....

WDove= V/2·(1-p)

Equating the two fitnesses, Hawk and Dove

V*(1-p)+(V/2-C/2)*p= V/2*(1-p)

... and solving for p

p= V/C

so for this "static point" where the Population Percent is an ESS solves to be ESS(percent Hawk)=V/C

TRY SYMBOLS

Taking the PAYOFF MATRIX results and plugging them into the above equation:

WHawk= V·(1-p)+(V/2-C/2)·p

Similarly for a Dove:

WDove= V/2·(1-p)+0·(1-p)

so....

WDove= V/2*(1-p)

Equating the two fitnesses, Hawk and Dove

V·(1-p)+(V/2-C/2)·p= V/2·(1-p)

... and solving for p

p= V/C

so for this "static point" where the Population Percent is an ESS solves to be ESS_{(percent Hawk)}=V/C

Multi image

Competive Coevolution - The Rough Skin Newt (Tarricha granulosa) has enough poison in its body to kill a human being 30 times over. This excess of lethal capability is due to the fact that it is involved in an arms race with a specific predator, the common garter snake, which in response to the newts poisonous defenses has over time has evolved to be highly tolerant of the poison. The two species are thereby locked in a "Red Queen" arms race.

Mutualistic Coevolution - Darwins orchid(Angraecum sesquipedale)and the moth Morgan's Sphinx (Xanthopan morgani) like many insect and flower "partners" have a mutual relationship where the moth gains pollen and the flower pollination. To insure efficiency in this particular exchange the pair have evolved a mechanism which excludes the waste of pollen transfer to/from another flower species and assures feeding pollin only to the "proper" pollinator. The mechanism is an extraordinarily long proboscis on the moth and a equally long neckery on the orchid