Stochastic logistic model with environmental noise

Brian Dennis (February 23, 2011)

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Abstract

The logistic model is the quintessential model of invasion. The fundamental idea of the logistic is "something replacing something else": vegetative cover replacing empty space, biomass replacing nutrient, infected individuals replacing uninfected ones, DVD households replacing VHS households. Biological mechanisms producing logistic growth are analogous to an autocatalytic chemical reaction converting a substrate to a product.

Given data on the abundance of a growing population, ecologists have handled the statistics of fitting the logistic model in a variety of ad hoc ways. Ecology textbooks are largely silent about estimating the unknown quantities in the logistic. In fact, different sources of variability in the data, that is, sources of departures of data from a logistic-predicted trajectory, would lead by statistical principles to different statistical methods for estimating parameters and predicting future outcomes.

Here I examine a particular stochastic version of the logistic model. The version is a diffusion process with environmental-type noise. The equilibrium (carrying capacity) is no longer a point equilibrium but rather is a gamma probability distribution. Many statistical properties of the model can be derived as formulas. With simulations, I evaluate an approximation, based on singular perturbation, for the full transition probability distribution of the process. The approximation turns out to be accurate and quite helpful for fitting the model to time series data. With the transition distribution in hand, incorporating sampling variability and estimating parameters with data cloning is straightforward. The model has the convenient property that the time intervals between observations can be unequal. Various examples that use the model for statistical analysis of population time series are presented.