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

Appendix A. The population projection matrix model.

The within-patch transition probability matrix (W_i) consists of seven age/stage classes (egg, four larval classes, pupal, and adult).  The parameters S are transition probabilities from the respective classes.  The parameter is the probability of post-natal dispersal out of the natal patch and _j is the probability of dispersal out of a patch by an adult greater than 25 d old.

The matrix for inter-patch movement of C. fenestrata from patch j to patch i is represented by T_ij. 

All inter-patch dispersal is assumed to be by adults, either by post-natal dispersers (represented by ) or by adults greater than 25 d old (represented by ).  The parameter is the probability of immigration from patch j to patch i on the condition that the beetle emigrated from patch j.  There are 5,550 matrices for inter-patch movement among the 75 patches.

The 75 matrices for intra-patch transitions and 5,550 matrices for inter-patch movement were assembled into one matrix (M).

The metapopulation of C. fenestrata is composed of 75 patches and each patch has 7 beetle stages, thus, M is a 525 by 525 cell matrix.  A 525-cell vector (N) represents the number of each stage of C. fenestrata in each patch.  The dynamic is a Markovian process where the population vector at time t+1 is a function solely of the population vector at time t;

(A.1)

In this study there are two alternate forms of matrix M.  M_nf applies to time periods without a flood, and M_f is for flood periods.  In the within-patch matrices (W's) in M_nf, the survival probabilities (S_E = 0.18, S_L = 0.87, S_P = 0.68, and S_A = 0.73) and reproductive rate (R = 4.4) are based on a combination of survivalorship and stage duration information (Johnson 2004a) and on the assumption that the population growth rate equals 1.  Parameter values within and outside the flood zone are identical during non-flood periods.  The probabilities that adult beetles immigrate and emigrate among patches, represented by the dispersal parameters were estimated in a concurrent study (Table 1; (Johnson 2003)).  The probability of natal dispersal (), that is dispersal by newly eclosed beetles, was set at 0 for the initial parameterization of the model.

The flood matrix M_f differs from M_nf in the following ways.  During a flood event all immature transition probabilities (S_E, S_L, and S_P) in the 17 flood zone patches were set at 0, thus, all immature beetles were assumed to be killed by flooding.  This is consistent with data on larvae (Johnson 2004b), but is an assumption regarding the eggs and pupae (the sensitivity of the model predictions to this assumption was analyzed and is in Appendix C).  Also, during a flood event, immigration was restricted to patches in the non-flooded zone.  Lastly, adult survival (S_A = 0.11 ± 0.07 SE) and emigration probabilities ( = 0.76 ± 0.02 SE) in flood zone patches during a flood were set at values estimated in Johnson (2004b).

LITERATURE CITED

Johnson, D. M. 2003. Spatial analyses of a Neotropical beetle: Cephaloleia fenestrata (Hispinae: Chrysomelidae: Coleoptera). Dissertation. University of Miami, Coral Gables, Florida, USA.

Johnson, D. M. 2004a. Life history and demography of Cephaloleia fenestrata (Hispinae: Chrysomelidae: Coleoptera). Biotropica 36:352–361.

Johnson, D. M. 2004b. Source-sink dynamics in a temporally heterogeneous environment. Ecology 85:2037–2045.