Appendix A. Selecting the appropriate basic incidence function model.

The incidence function model (IFM) makes specific assumptions about various dynamics in a system. To critically determine an appropriate form for the patch occupancy model to use for the *C. fenestrata *system, I parameterized forms of the model (using simulated annealing) with various assumptions. Because testing every possible permutation of the model would have involved 128 models, I chose instead to test models in a step-wise fashion, selecting the model that best fits the data with the minimum number of parameters for each of the following functional forms. For the dispersal kernel, I tested a power, exponential, Bessel K, and Gaussian decay function. The model was tested with and without the Allee effect on colonization probability. Two functional forms of the extinction-patch size relationship were tested: power and exponential decay. A previous study indicated that *C. fenestrata* has a ‘post-natal dispersal’ phase that significantly exceeds that of older adults (Johnson and Horvitz 2005), thus, I added post-natal dispersal to one of the modified models. Lastly, it is unclear whether immigrants from outside of the study area have a significant effect on the regional dyanamics of *C. fenestrata*, so I added ‘remote migration’ to the model as a model-estimated parameter in one of the forms. Each grouping of rows in the table is a comparison of different modifications in each of the above categories. The model with ΔAIC = 0.00 was selected as the best fit model and then transferred to the next below grouping of model modifications (except in the case of postnatal dispersal and remote migration, in which cases the simpler models were selected as the most parsimonious models because the ΔAIC values were quite low). These analyses indicated that the best fit and most parsimonious model was of the following form: (1) with a power decay function of dispersal, (2) without an Allee effect, (3) with a power decay function for the extinction-patch size relationship, (4) without a rescue effect, (5) without post-natal dispersal, and (6) without remote migration.

Disp |
Allee |
Extinction |
Rescue |
PN Disp |
R. Migr |
AIC |
ΔAIC |

Comparing dispersal kernels |
|||||||

P |
Y |
P |
Y |
N |
N |
437.42 |
0.00 |

B |
Y |
P |
Y |
N |
N |
462.56 |
25.14 |

G |
Y |
P |
Y |
N |
N |
466.32 |
28.90 |

E |
Y |
P |
Y |
N |
N |
466.98 |
29.56 |

Comparing models with and without Allee effect |
|||||||

P |
N |
P |
Y |
N |
N |
378.52 |
0.00 |

P |
Y |
P |
Y |
N |
N |
437.42 |
59.90 |

Comparing extinction probability functions |
|||||||

P |
N |
P |
Y |
N |
N |
378.52 |
0.00 |

P |
N |
E |
Y |
N |
N |
382.24 |
3.72 |

Comparing models with and without a rescue effect |
|||||||

P |
N |
P |
N |
N |
N |
362.58 |
0.00 |

P |
N |
P |
Y |
N |
N |
378.52 |
15.94 |

Comparing models with and without postnatal dispersal |
|||||||

P |
N |
P |
N |
N |
N |
362.58 |
0.00 |

P |
N |
P |
N |
Y |
N |
362.58 |
0.00 |

Comparing models with and without remote migration |
|||||||

P |
N |
P |
N |
N |
Y |
359.84 |
0.00 |

P |
N |
P |
N |
N |
N |
362.58 |
2.74 |

Disp = decay function of the dispersal kernel (P = power, E = exponential, B = Bessel K, and G = Gaussian), Allee = Allee effect on colonization probability (Y = yes and N = no), Extinction = functional form of the extinction-patch size relationship (P = power and E = exponential), Rescue = rescue effect (Y = yes and N = no), PN Disp = postnatal dispersal as measured in Johnson and Horvitz (2005) (Y = yes and N = no), R Migr = remote immigration from outside the study area (Y = yes and N = no), AIC = Akaike’s information criterion, and ΔAIC the difference between the lowest AIC and that of the current model. A general rule of thumb is that ΔAIC values < 2 means that one is unable to distinguish between two models, 2 < ΔAIC < 7 means it is suggestive but that the two models are not reliably distinguishable. The models in the top four rows differ only in the dispersal decay functions. The power function of dispersal had strong empirical support because it had the lowest AIC value (ΔAICs > 25), thus, the power function was selected for the dispersal kernel. When the Allee effect was removed from the model this greatly improved the model fit (ΔAIC = 59.90). The exponential extinction function was then added to the model, but this was a weakly poorer fit than the model with a power decay extinction function (ΔAIC = 3.72), thus, suggesting that either model may describe beetle movement equally well, but evidence for the power decay function was slightly stronger. The model lacking the rescue effect was a strongly better fit than the model with the rescue effect (ΔAIC = 15.94). Adding postnatal dispersal did not improve the model fit (ΔAIC = 0.00), thus, the simpler model (without postnatal dispersal was selected as the most parsimonious model. Lastly, adding remote migration to the model only marginally improved the model fit (ΔAIC = 2.74). This low ΔAIC indicates that the fits of the two models are rather indistinguishable, thus, the model without remote dispersal was selected because it was the most parsimonious.

LITERATURE CITED

Johnson, D. M., and C. C. Horvitz. 2005. Estimating postnatal dispersal: tracking the unseen dispersers. Ecology **86**:1185–1190.