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Modelling fungal colony counts
One way of estimating concentrations of viable fungal spores in soil is to mix a measured amount of soil with distilled water, pour some of the mix onto agar plates, incubate these for a period of time and then count the number of colonies that are formed on the plate. When multiplied by an appropriate conversion factor, this gives an estimate of the number of colony forming units per gram of soil.
In principle, when using this method, the average number of colonies per plate should be proportional to the amount of soil per plate. But in practice, it is common to find that the number of colonies grows more slowly than linearly when the amount of soil is increased. This can occur for various reasons. One possibility is that colonies that grow from spores that are close together on the plate coalesce and appear as a single colony. Another possibility is that some spores are inhibited from growing by chemicals or other inhibitory organisms that are present in the soil suspension.
Various analyses of colony counts of the organism Verticillium dahliae, which causes wilt in many crops, are published in two papers.
The first [Ref 1] considers various aspects of design and analysis in relation to a particular non-linear model. Whilst this model can be derived as an approximate model arising from overlapping colonies, the parameter estimates that we obtain are very different to those that would be implied by the physical dimensions of the colonies and in this application the model is best considered simply as an empirical model that provides a good fit to the data.
The second paper [Ref 2] presents a model of inhibition in which colonies fail to develop if spores are close to spores of other inhibitory species. The model has three parameters, but a limiting case of the model with only two parameters is shown to be more useful in practice. This latter model, which is a generalized linear model, is fitted to the V. dahliae counts and the model is contrasted with the earlier model.
References
[1] Morris D.L. & Ridout M.S. (2005) Models for microbiological colony counts. Journal of Agricultural, Biological & Environmental Statistics, 10, 158-169. [journal link]
[2] Ridout M.S. & Harris D.C. (1997) Estimating the concentration of propagules of a plant pathogen from soil dilution plate counts when the response is nonlinear. Applied Statistics, 46, 111-121. [journal link]