The G-Function Method for Analyzing Darwinian Dynamics

Darwinian dynamics refers to the dynamical processes underlying natural selection that drives evolution. We are interested in the evolution of strategies used by biological entities. There are two dynamical processes involved, population dynamics (relationship between population density and the agents affecting density) and strategy dynamics (relationship between strategy values and the agents affecting these values). Darwinian dynamics is a total dynamic obtained through the coupling of these two processes, the modeling of which, involves dynamical systems, optimization, stability, and game theory. Using a method called the G-function approach, we explore how an evolutionary process can take place in a set of differential equations, and we examine some interesting links between evolutionary stability and optimization as embodied in the ESS maximum principle. One of the interesting paradoxes is how a "hill-climbing" algorithm can end up at a stable local minimum and why this might have important implications in understanding speciation (the creation of new species from a homogeneous population). Finally, we will examine how these concepts are currently being applied to model the development of tumors in humans.

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