The distribution of an organism species in the environment deviates frequently from randomness due to natural cycles, availability of food resources and avoidance of harm. As a result, observed data can show over-dispersion, zero-inflation and even heavy tail. Models such as the negative binomial (NB), Poisson-inverse Gaussian (PIG), and zero-inflated Poisson are frequently used in applications instead of the Poisson distribution which is usually the default model. This paper uses a three-parameter discrete distribution that unifies distributions such as Poisson, NB, PIG, Neyman Type A, and Poisson-Pascal. The three-parameter family covers a wide range of tail heaviness relative to NB, and thus suitable for modelling over-dispersed count data with a shorter or longer tail. Moreover, it shows some capacity for zero-inflated data. Grouped counts of coliform bacteria from Lake Erie and counts of European corn borer larvae in field corn are used to illustrate the application of the model and the associated likelihood-based inferences. Copyright © 2010 John Wiley & Sons, Ltd.