Contribution of aij’s
Contributions of paired matrix elements to V(l) have been traditionally presented in three dimensions (Fig. B.1). Elements along the diagonal represent the variance of a transition value, aij, (Cov(aij, aij)); all other elements represent the covariance among a pair of transitions (aij vs. akl) over the eight neighbor-interaction/census matrices. The highest peaks are associated with the variance among stasis for reproductive size classes and the covariance values among these transition probabilities. Therefore, stasis of reproductive size classes make the largest contribution to the variation in l observed among matrices. Minor peaks in the opposite corner of the diagram represent variances among growth of new individuals (a31), recruitment of the largest juvenile size class (a43), and the covariance among these two transitions.
Cij, which gives the contribution of individual matrix entries, shows that stasis of reproductive individuals accounts for most of V(l) (Fig. B.2). Although l is lower for all neighbor-interaction/census matrices when compared to A(..), negative contributions from covariances among matrix entries are relatively insignificant.
Contributions of neighbors and census
Examination of the relative effects of neighbor-interaction and census treatments to variation in l show that although neighbor effects yield a larger reduction in population growth than census effects, there was greater variation among censuses than among neighbor-interaction type (Fig. B.3). Interaction effects show variation in the effects of neighbors from 1984-1989 compared to 1989-1994. Neighbors give a slightly less negative contribution to l during 1984-1989 while giving a slightly more negative contribution from 1989-1994.
We can further decompose the treatment effects by observing the differences between each individual transition (aij) of the treatment matrix and ten-year reference matrix, A(..), and compare these differences to the contribution of each aij to the difference in l. Inspection of neighbor effects shows that large negative differences occurred between transitions that involved the fecundity of reproductive size classes (a14, a15, a16) (Fig. B.4 a,b). However, these differences make both positive and negative contributions to the observed differences in l (Fig. B.4 c,d). Large differences did not occur between transitions involving the juvenile size classes of the neighbor-defined matrices and A(..), nor did these transitions contribute greatly to the observed differences in l. Therefore, the differences in population growth rate between neighbor-defined ‘patches’ and the ten-year reference matrix were largely due to the dynamics among adult individuals.
Effects among censuses show contrasting results. During the 1984-1989 census period, most differences between transitions of the treatment and reference matrix occur among transitions involving juvenile size classes. Both small positive and small negative differences occur (Fig. 9a). In contrast, very small differences among transition values of the census and ten-year reference matrix occur among the smaller size classes during the 1989-1994 census period, but there are large negative differences among transitions of these two matrices involving fecundity of reproductive size classes (Fig. 9b). These differences reflect juvenile transitions yielding a negative contribution to l during 1984-1989, with peaks associated with the growth of small juveniles (a31) and recruitment of large juveniles (a43), and adults yielding a smaller net positive contribution (Fig. 9c). During 1989-1994, the opposite was true. Juveniles contributed positively to l, with similar peaks for growth of small juveniles and recruitment of large juveniles, while adults had a greater negative contribution with definite peaks corresponding to fecundity (transitions a14, a15, a16) (Fig. 9d).
In short, juvenile dynamics tended to decrease population growth during 1984-1989 and to increase population growth rates during 1989-1994. In contrast, the dynamics among adults tended to increase population growth rates during 1984-1989 and to decrease population growth during 1989-1994.
Observation of the interaction between neighbor-interaction and census show that all neighbor-interaction/census matrices show large negative differences among transitions involving fecundity (transitions a14, a15, a16) when compared to our ten-year reference matrix (Fig. B.5). Positive and negative fluctuations occur among transitions involving the juvenile size classes among all neighbor-interaction/census matrices and the ten-year reference matrix to varying degrees. The contribution of these differences to observed differences in l does not suggest an interaction between the effects of neighbors and census involving the juvenile size classes. Neighbors led to negative contributions of transitions among juvenile size classes during 1984-1989, but positive contributions from juveniles from 1989-1994 with peaks corresponding to growth and recruitment. The same pattern occurred for isolated individuals. Juvenile size classes yielded a negative contribution from 1984-1989, but yielded a positive contribution from 1989-1994, again with peaks corresponding to growth and recruitment. Reproductive size classes gave smaller positive and greater negative contributions to l for all neighbor-interaction/census combinations.