Ecological Archives M078-019-A5

C. Wirth, J. W. Lichstein, J. Dushoff, A. Chen, and F. S. Chapin, III. 2008. White spruce meets black spruce: dispersal, postfire establishment, and growth in a warming climate. Ecological Monographs 78:489–505.

Appendix E. Derivation of negative binomial distribution for the number of recruits.

We assume that seed rain (S) comes from a negative binomial distribution with parameters q (probability of success) and r (shape), so that the probability of s seeds arriving is

EqE1

with mean EqE2 and variance EqE3 (Casella and Berger 2002). We assume that each seed has a probability p of establishing and surviving, so that the number of recruits (N) comes from a binomial distribution with s trials and probability of success p . The conditional probability of n seeds recruiting given seed rain s is then

EqE4

The unconditional, or marginal, probability of n seeds recruiting is

EqE5

where us - n and α ≡ 1 - (1 - p)(1 - q). The summand is a negative binomial probability, so

EqE6

which is negative binomial with mean EqE7 and variance EqE8.

LITERATURE CITED

Casella, G., and R. L. Berger. 2002. Statistical Inference. Second Edition. Duxbury Press. 688 pp.


[Back to M078-019]