gcb2522-sup-0001-appendixS1.pdfapplication/PDF, 199K | **Appendix S1.** Detailed methods. |

gcb2522-sup-0002-tables1.xlsapplication/msexcel, 51K | **Table S1.** Model selection. Study site indicates the study site (Puerto Rico or Thailand), Model indicates the density-dependence model (Beverton-Holt, Gompertz or Ricker), Stochasticity indicates whether the model had Observation or Environmental stochasticity, Form indicates whether covariates were considered linearly (Linear) or nonlinearly (Threshold, etc.), − loglik is the negative loglikelihood of the model, Parameters is the number of parameters consider by the model, AIC is the Akaike Information criterion for the models, BIC the Bayes information criterion for the model, and ∆AIC & ∆BIC the difference with respect to the minimum value. |

gcb2522-sup-0003-figures1.pdfapplication/PDF, 145K | **Figure S1.** Seasonality of the studied periods and background seasonality. Rainfall in (a) Puerto Rico, (b) Thailand, Temperature range in (c) Puerto Rico, (d) Thailand. Boxplots represent the seasonal distribution of climatic variables (estimated with data from 1951 to 2011). Blue and red lines represent the records from our study periods for rainfall and temperature range, respectively. For the estimates, we used freely available monthly data from the US National Oceanic and Atmospheric Administration, NOAA (ftp://ftp.ncdc.noaa.gov/pub/data/ghcn/v2/) for San Juan, Puerto Rico (Station 435785260), and for Thailand, we used data from Bangkok (Station 228484550, the closest location to Chachoengsao with long climatic records). For Thailand, it is worth noticing that temperature data from 1972 to 1995 were missing, the time period including the study period at Chachoengsao (1990–1993). In general, climatic patterns for our study periods were encompassed by the overall seasonal variability of the study sites. However, February & March 1993 in Puerto Rico, and March & April in 1992 in Thailand had an extreme temperature range when compared with their long-term seasonal profile. |

gcb2522-sup-0004-figures2.pdfapplication/PDF, 25K | **Figure S2.** Time domain descriptive analysis of mosquito density *N*_{t}. Autocorrelation Function, ACF, for Puerto Rico (a) and Thailand (c); Partial Autocorrelation Function, PACF, for Puerto Rico (b) and Thailand (d); Cross correlation functions between *N*_{t} and: rainfall in Puerto Rico (e) and Thailand (g) Average temperature in Puerto Rico (f) and Thailand (h); maximum temperature in Puerto Rico (i) and Thailand (k), and minimum temperature in Puerto Rico (j) and Thailand (l). Blue dashed lines indicate 95% confidence intervals of correlation expected by random, i.e., peaks outside the band indicate a significant correlation. The *x*-axis indicates time lags (in weeks) and the *y*-axis are correlation values [i.e., contained in (−1,1)]. |

gcb2522-sup-0005-figures3.pdfapplication/PDF, 15K | **Figure S3.** Time frequency domain descriptive analysis. Cross-wavelet coherency and phase of mosquito abundance in Puerto Rico with (a) rainfall, (b) average temperature, (c) maximum temperature, and (d) minimum temperature; and in Thailand with (e) rainfall, (f) average temperature, (g) maximum temperature, and (h) minimum temperature. In each panel, the top plot shows coherency and the bottom plot shows the phase. The coherency scale is from zero (blue) to one (red). Red regions in the upper part of the plots indicate frequencies and times for which the two series share variability. The cone of influence (within which results are not influenced by the edges of the data) and the significant (*P* < 0.05) coherent time–frequency regions are indicated by solid lines. The colors in the phase plots correspond to different lags between the variability in the two series for a given time and frequency, measured in angles from −PI to PI. A value of PI corresponds to a lag of 26 weeks. A smoothing window of 26 weeks (2 w + 1 = 53) was used to compute the cross-wavelet coherence. |

gcb2522-sup-0006-figures4.pdfapplication/PDF, 10324K | **Figure S4.** Model fitting, fitted N̂t vs observed *N*_{t} mosquito densities for the best autonomous models for Puerto Rico (a) and Thailand (b) and the best forced models for Puerto Rico (c) and Thailand (d). r̂ indicates the estimated Pearson correlation between N̂t and Nt for the model presented in each panel. |

gcb2522-sup-0007-figures5.pdfapplication/PDF, 293K | **Figure S5.** Model Simulation. Summary of 1000 simulations for the best autonomous models for Puerto Rico (a) and Thailand (b), and the best forced models for Puerto Rico (c) and Thailand (d). For all the iterations (a.k.a., time steps) of each model, we computed the 95% confidence limits (blue lines) and the median (green lines) of the simulations. Red lines represent sample simulations. Open circles represent the population density (i.e., observed data) in panels for autonomous models (i.e., a and b) and the population density with the forcing subtracted (DFR in Fig. of the main text) in panels for forced models (i.e., c and d). In the forced models, we show DFR, because for the simulations, we excluded the impact of the forcing. For the simulations, we employed parameter estimates from Table and equation (1) from the main text to generate realizations of the best autonomous and forced models. We used as initial conditions, the first and second observations of the observed time series for the Puerto Rico models (107 realizations per simulation) and observations nine and 10 of the observed time series for the Thailand models (143 realizations per simulation). This figure shows that, in general, simulations of the autonomous models were unable to capture the most extreme values of the observed data. |

gcb2522-sup-0008-figures6.pdfapplication/PDF, 17K | **Figure S6.** Bootstrap of the stability. Parametric bootstraps to test the stability of the Gompertz model with parameter estimates for the best autonomous (i.e., no forcing) models of (a) Puerto Rico, (b) Thailand, and the best forced models of (c) Puerto Rico and (d) Thailand. Plots present the distribution of the largest eigenvalues (*ξ*) from the jacobian of 10 000 simulations of the corresponding Gompertz model assuming neutral stability (i.e., the distribution is expected to have a mean of 1). The *y*-axis is the probability density for a given value of *ξ*, i.e., the *x*-axis. The dashed lines represent the estimated dominant eigenvalues ξ̂ of the jacobian evaluated at the nontrivial equilibrium using parameter estimates of the best models presented in Table . In all four cases, the null hypothesis that stability is neutral (i.e., not different from 1) is rejected (*P*<0.05, *P*-values for the specific models are presented in each panel), as ξ̂ is significantly smaller than 1 in each case. These results indicate that models are stable, because the largest eigenvalue for each model is contained within the unit circle, the stability criterion for discrete time models (Levins & Wilson, 1980). |