continuous or discrete time. Continuous time is modelled using differential equations. Discrete time is modelled using difference equations. These model ecological processes that can be described as occurring over discrete time steps. Matrix algebra is often used to investigate the evolution of age-structured or stage-structured populations. The Leslie matrix, for example, mathematically represents the discrete time change of an age structured population. Models are often used to describe real ecological reproduction processes of single or multiple species. These can be modelled using stochastic branching processes. Examples are the dynamics of interacting populations (predation competition and mutualism), which, depending on the species of interest, may best be modeled over either continuous or discrete time. Other examples of such models may be found in the field of mathematical epidemiology where the dynamic relationships that are to be modeled are host–pathogen interactions. Bifurcation theory is used to illustrate how small changes in parameter values can give rise to dramatically different long run outcomes, a mathematical fact that may be used to explain drastic ecological differences that come about in qualitatively very similar systems. Logistic maps are polynomial mappings, and are often cited as providing archetypal examples of how chaotic behaviour can arise from very simple non-linear dynamical equations. The maps were popularized in a seminal 1976 paper by the theoretical ecologist Robert May. The difference equation is intended to capture the two effects of reproduction and starvation. In 1930, R.A. Fisher published his classic The Genetical Theory of Natural Selection, which introduced the idea that frequency-dependent fitness brings a strategic aspect to evolution, where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable offspring. In 1961, Richard Lewontin applied game theory to evolutionary biology in his Evolution and the Theory of Games, followed closely by John Maynard Smith, who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting", defined the concept of the evolutionarily stable strategy. Because ecological systems are typically nonlinear, they often cannot be solved analytically and in order to obtain sensible results, nonlinear, stochastic and computational techniques must be used. One class of computational models that is becoming increasingly popular are the agent-based models. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in