concepts. Deterministic models (systems of differential or difference equations) dominated theoretical advances for much of the history of the field, and they continue to be the single most important choice of modeling framework for analytical models. In the 1920s and 1930s, two key concepts were formalized using deterministic models: competition and predation. Mathematical models greatly enhanced our understanding of both processes. Competition has been identified as an important process of ecological communities ever since Darwin proposed it as the chief mechanism in the evolution of species (Darwin, 1859). The competition models by Lotka (1932) and Volterra (1926), formulated as systems of differential equations, provide a theoretical framework for the dynamic interactions within a trophic level.1 This framework was further developed by Elton (1927, 1933) using the concept of a niche, which he defined as “the status of an animal in its community.” He linked this concept to competition in order to explain how multiple species can persist within a community. A mathematical formulation of the niche concept was finally given by Hutchinson (1957), who defined a niche as a subset of an n-dimensional hypervolume. This concept is still useful today. While the models of Lotka and Volterra describe phenomena, they lack mechanisms for competition. Tilman’s (1982) resource competition model led the way from phenomenological to mechanistic competition models. Like the Lotka-Volterra models, mechanistic competition models are also based on systems of differential equations and continue to form the conceptual basis for understanding competition among multiple species.

Predation is by definition a process that occurs between trophic levels. Lotka (1925) and Volterra (1926) were the first to provide a mathematical formulation of this process, again using systems of differential equations. Differential equations model continuous time dynamics and are thus well suited for populations with overlapping generations. However, this does not always hold for biological situations. For instance, the seasonal dynamics of a host and an associated parasitoid2 are better described by discrete time models. To include this aspect of biological realism into models, Nicholson and Bailey (Nicholson, 1933; Nicholson and Bailey, 1935) promoted systems of difference equations to describe predation models. Difference equations are now commonly employed to model interactions among species with nonoverlapping generations.

In the 1950s and 1960s the focus shifted toward understanding the

1  

A trophic level is a stratum of the food chain consisting of species the same number of steps from the primary source of nutrition.

2  

A parasitoid is an insect that lays its eggs in, on, or near a host and whose offspring consume the host as they develop.



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