Appendix A. Comparison of the model output for type II and type III functional responses.
The type III response model corresponding to that in Dwyer et al. (2004) but with sequential action by pathogens and predators would be:
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(A.1) |
in place of Eq. 7 of the main text; all parameters and variables are as defined in the main text. Overall The type II and type III models behave similarly. The type III model response also result in a decrease of outbreak frequency with enhanced predation pressure (Fig. A1). However, the effect was slightly less pronounced and the gypsy moth population was predicted to go extinct more rapidly with the increase of the predator carrying capacity (compare Fig. A1 with Fig. 6).
At a relatively high predator level (K = 7), the same dominant period of around 7-9 years appeared for both models (Fig. A2). Superharmonic periods were more pronounced in the case of the type II functional response and the overall amplitudes were greater.
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FIG. A1. Phase portraits representing pathogen density vs. host density (log10-transformed), with predators set to a constant ( = 0) (a) K = 2, (b) K = 7. We made 50 simulations for each couple of initial values and we plotted the last 30 points. We used the same initial values as Dwyer et al. (2004) for their Fig. 3b. |
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| FIG. A2. Periodograms for type III (a) and type II (b) functional responses with a constant predator level (K = 7). |
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| FIG. A3. Examples of realizations of the deterministic two-patch model with varying strength of spatial coupling. Predator carrying capacities are assumed constant at K = 6 (patch 1 – red lines and circles) and K = 1 (patch 2 – black lines) and movement rates (d) assumed to vary. Associated power spectra are given in Fig. 7 in the main text. |
Literature cited
Dwyer, G., J. Dushoff, and S. H. Yee. 2004. The combined effects of pathogens and predators on insect outbreaks. Nature 430:341–345.