Geometry, heterogeneity and competition in variable environments

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Abstract
The effects of environmental fluctuations on coexistence of competing species can be understood by a new geometric analysis. This analysis shows how a species at low density gains an average growth rate advantage when the environment fluctuates and all species have growth rates of the particular geometric form called subadditive. This low density advantage opposes competitive exclusion. Additive growth rates confer no such low density advantage, while superadditive growth rates promote competitive exclusion. Growth-rate geometry can be understood in terms of heterogeneity within populations. Total population growth is divided into different components, such as may be contributed by different life-history stages, phenotypes, or subpopulations in different microhabitats. The relevant aspects of such within-population heterogeneity can be displayed as a scatter plot of sensitivities of different components of population growth to environmental and competitive factors, and can be measured quantitatively as a covariance. A three-factor model aids the conceptual division of population growth into suitable components.