Karen E. Rose, Svata M. Louda, and Mark Rees. 2005. Demographic and evolutionary impacts of native and invasive insect herbivores on Cirsium canescens. Ecology 86:453–465.

Appendix A. A description of the negative binomial distribution for aggregated oviposition by Rhinocyllus conicus on flower heads of Platte thistle.

The distribution of Rhinocyllus eggs in the population can be described using a negative binomial distribution which is characterised by two parameters: the mean number of eggs per plant, , and a clumping parameter,. The probability that e eggs are laid on a plant,, is given by the coefficients of the power series expansion of the generating function, , which is defined by

As  the distribution approaches a Poisson distribution, which implies that each egg is laid independently at random. Seed production can be written as

where x is root crown diameter and e the number of  Rhinocyllus eggs, see equation (4); the effect of native insects has for simplicity been subsumed into the parameter A. Define  then using generating function, , gives,

          .

But the mean number of eggs depends on plant size, so . See Table 1 for parameter estimates.