Analysis of Steady‐State Populations With the Gamma Abundance Model: Application to Tribolium

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Abstract
We develop a stochastic model for the abundance of adult populations of the four beetle Tribolium. The model is in the form of a stochastic differential equation containing adult recruitment and mortality rates perturbed by multiplicative noise. A deterministic version of the model (an ordinary differential equation) predicts a fixed, stable equilibrium; by contrast, the stochastic model predicts a stationary probability distribution for population size. The model can be approximated closely by a stochastic logistic equation having a gamma distribution as a stationary solution. We develop or clarify various practical aspects of making statistical inferences for the resulting gamma abundance model, including estimation of parameters, testing goodness of fit, obtaining confidence intervals for functions of parameters, and testing to compare two gamma distributions. Analyses of 11 data sets on Tribolium suggest that the gamma model deserves more widespread consideration as an equilibrium abundance model for other species.