How a population develops and evolves in a given time period of interest is a fundamental research subject in evolutionary biology and ecology, with applications in many other fields of biology and medicine, ranging from virus infection to bacteria development, from plant succession to animal breeding, and from genetic disorder to cancer evolution. Early work on modeling evolution and natural selection traces back to R. Fisher’s seminal thesis on the fundamental theorems of natural selection completed almost a century ago. Later, differential equations such as the famous Lotka-Volterra equations were introduced to model population dynamics of various types. Evolutionary game theory was developed in 1970s when John Maynard Smith introduced the game theory to biology and proposed the concept of evolutionary stability for equilibrium states.
In evolutionary game theory, species are considered as if they are players in a game, competing for resources, for survival, and for reproduction. A game model can then be established for the study of any given population of competing species – for analysis of population changes and for prediction of equilibrium states and their stabilities. The theory involves such mathematical branches as game theory, optimization theory and methods, and ordinary differential equations, and extends further to graph theory and stochastic processes. It is still developing, though, with theoretical issues yet to be addressed, simulation methods to be tested, and important applications to be explored.