Appendix A. Derivation of model for sea lice and juvenile salmon population dynamics.
Assuming that the distribution of sea lice on juvenile pink salmon hosts, q(p), is Poisson yields a solution to the sum in Eq. A.1 given by
where the expectation of p is given by and the expectation of p2 is given by (Anderson & May 1978). Because the model for juvenile salmon mortality due to parasitism (either direct or via predators) is piece-wise linear, the approximation of the Poisson distribution for parasites on the host population is preserved (Rousset et al. 1996).
With equation (A.1), the model for sea lice and juvenile pink salmon population dynamics becomes
Finally, we further simplified the model by rewriting it in terms of the differential equation for mean parasite abundance (, average number of lice per fish) rather than total parasite abundance (P(t)) because it is a more biologically intuitive and informative quantity to track. By solving
we obtain the finalized model for sea lice and juvenile salmon population dynamics
Rousset, F., F. Thomas, T. deMeeus, and F. Renaud. 1996. Inference of parasite-induced host mortality from distributions of parasite loads. Ecology 77:2203–2211.