Evolutionary Games and Ecosystems (6)

Game Theory and Lotka-Volterra Equation

Zebras and Lions, then some Hyppos

Before proceeding, we have to cover something called "Lotka-Volterra system" which is one of the central concepts in ecology and quantitative biology. This features large in our arguments too, and we would like to give the readers without mathematical traing some taste of it by explanation solely in verbal manner.
Think of Savanna in Tanzania, where Zebras and Lions can be spotted. Azure sky, herds of black and white, heards of black and yellow. A massai worrier sits on top of a rock and drinks from wrest poach. He senses the afternoon rain from the odour of wind..
In fact, you can forget about Massai, for now.
Zebras survive by eating grass, so can be regarded as self-sustaing. Lions on the other hand, are unable to eat grasses, thus go extincted without Zebras, upon which they prey. Temporal change of the populations of Zebras and Lions are interreled, and that is mathematically described by the celebrated non-linear Lotka-Volterra equation. It has a single stable constant (time-invariant) solution, which underpins the coexistence of Zebras and Lions in savanna forming the master-slave relation.

Let us now move on from Tanzania to a savanna in neighboring country along the beautiful lake. No lions can be spotted here. Locals say this place once has been a part of Zulu kingdom under the famed conquirer King Shaka. It is said that Zulu brought in Hyppopotomus with them. But somehow, Zebras persisted while driving Hyppos into extinctions, and Zulus killed off all Lions.

In fact, in anyplace on earth, two species surviving on a common grassland do not coexitst, but one of them goes extinct. Situation is the same for two competing predator species surviving on a common prey.　The reason behind this is mathematical again: Except in the case of very week aggression intensity for both parties, Lotka-Volterra equation describing the time evolution of two competing species donot have any stable constant solution aprt from the 'trivial" ones with either population going zero. "No two heros coexist" is the ancient eastern proverbe for this. Cato senior's repeted call inn the Senate for the distruction of caltage could be also understood as a mathematical statement.

To the confused good readers, we stress that the point really was "Zebra and Lion". And also "Zebras and Hyppos". Other entrants to the scene, Shaka, Cato, Plato and Sun Tzu, could all be safely forgotten.

Representing Game Theory as Differential Equations

With the magic word "replicator dynamics", the evolution in the selection of optimal strategy in game theory can be turned into nonlinear differential equation named Lotka-Volterra equation. Nash equilibrium of a game corresponds to the stable constant solution of the Lokta-Volterra equation. Actual calculation of mixed Nash (including the one for Devil-Angle game) could be easier in this differential equation from.
But we are now left with too little space probably because of the descrption of colors in savanna. We shall change tha chapter.