Online Resources for: Long-term shifts in the cyclicity of outbreaks of a forest-defoliating insect
Andrew J. Allstadt1, Kyle J. Haynes1, Andrew M. Liebhold2, and Derek M. Johnson3
1 The Blandy Experimental Farm, University of Virginia, 400 Blandy Farm Lane, Boyce, Virginia 22620, USA
2 USDA Forest Service, Northern Research Station, 180 Canfield Street, Morgantown, WV 26505, USA
3 Department of Biology, University of Louisiana, Lafayette, Louisiana, 70504, USA
Online Resource 1: Correction for bias present in the computation of wavelet coefficients
Liu et al. (2007) demonstrated that the traditional method of computing wavelet power coefficients is biased towards producing higher power values for low-frequency (long period) signals. Power has traditionally been confused with the power integrated across time. Since the range of integration increases with the wavelet scale, power integrated across time tends to be greater at larger scales (Liu et al. 2007). To correct for this bias, we follow the protocol recommended by Liu et al. (2007), which is to divide the integrated wavelet power by the wavelet scale.
Online Resource 2: Additional material from the gypsy moth population model
Selection of predation functional response curves
As described in the main text, we parameterized multiple functional response curves (Eq. 4) incorporating the fluctuating predator density model (Eq. 6) and characteristics of the functional response from previous versions of the gypsy moth population model. By retaining the predation half saturation point (N'=0.05; Dwyer et al. 2004) and following Haynes et al. (2012), we could calculate parameters c and d given a known fraction of pupae killed as Nt'→0 (F0), and the predator density at that time (PF). Bjørnstad et al. (2010) used F0=0.98, but the value of PF was uncertain. Therefore, we ran the fluctuating predator density model for 106 time steps, and parameterized multiple functional responses by setting PF equal to various quantiles of the resulting predator density distribution. We determined which functional responses produced gypsy moth population dynamics similar to that found in nature. Specifically, we looked at patterns of gypsy moth mortality through time, expecting that predation should be a major source of mortality when the gypsy moth is at low densities (Campbell and Sloan 1977; Elkinton et al. 1996), and that high density outbreaks are brought to an end by high mortality due to disease (Leonard 1981; Dwyer et al. 2000). We found that using functional responses based on quantiles less than 0.7 led to mortality dominated by predation at all times (Fig. OR1a), i.e. predation was too strong. On the other hand, functional responses based on quantiles greater than 0.9 led to mortality solely due to disease (Fig. OR1c). Quantiles in the range [0.7, 0.9] produced patterns of mortality similar to those in nature (Fig. OR1b). As a result, all subsequent simulations were carried out using the functional response based on the quantile of 0.8, where PF = 12.25, c = 0.3195, d = 0.0454 and strong Allee effects could occur only if Pt 13.5.