Statistical Models for Limiting Nutrient Relations in Inland Waters

Abstract

The ecological theory of limiting factors holds that the observed level of response in a biological process will be governed by the input factor in least supply—the limiting factor. This theory has formed the basis for numerous attempts by aquatic ecologists to describe the relation between the biological productivity of inland waters and the availability of plant nutrients required for algal growth. Regression analysis has been the primary statistical tool used in the development of such relations, yet any statistical model that represents the limiting effect of some explanatory factor as an expectation contradicts the substantive theory of limiting factors. Limnological data not resulting in an adequate regression of chlorophyll on phosphorus have been viewed as failing to support the limiting effect of this nutrient on algal biomass in lakes. But when represented by a more appropriate model, such data may be seen to provide similar evidence for the relation of chlorophyll to phosphorus as does data resulting in a strong regression. Data from limnological studies often exhibit a scatter of points distributed in the shape of a triangle lying beneath an upper boundary. Appropriate models for such data are introduced to describe the upper boundary or potential limit, the distribution of points falling below the limit, and the degree of random error. An application of the EM algorithm provides marginal maximum likelihood estimates of the parameters in the more complex models considered. Several results are given for the models, including a goodness-of-fit diagnostic and estimation of the large-sample parameter covariance matrix. Application of the models is illustrated by fitting empirical relationships between chlorophyll and the plant nutrient phosphorus in temperate lakes.

Key Words:

• EM algorithm
• Limit function
• Marginal maximum likelihood
• Mixture distributions