Inverse modeling for effective dispersal: Do we need tree size to estimate fecundity?

The estimation of the dispersal kernel for the seedling and sapling stages of the recruitment process was made possible through the application of inverse modeling to dispersal data. This method uses the spatial coordinates of adult trees and the counts of seedlings (or saplings) in small quadrats to estimate the dispersal kernel. The unknown number of recruits produced by an adult tree (the fecundity) is estimated - simultaneously with the dispersal kernel - via an allometric linear model relating the unknown quantity with a (easily) measured characteristic of the adult tree (usually the basal area). However, the allometric relation between tree size and reproductive success in the sapling (or seedling) stage may not be strong enough when numerous, well-documented, post-dispersal processes (such as safe-site limitation for recruitment) cause large post-dispersal seedling mortality, which is usually unrelated to the size of the tree that dispersed them. In this paper we hypothesize that when tree size and reproductive success in the seedling/sapling stage are not well correlated then the use of allometry in inverse modeling is counter-productive and may lead to poor model fits. For these special cases we suggest using a new model for effective dispersal that we term the unrestricted fecundity (UF) model that, contrary to allometric models, makes no assumptions on the fecundities; instead they are allowed to vary freely from one tree to another and even to be zero for trees that are reproductively inactive. Based on this model, we examine the hypothesis that when tree size and reproductive success are weakly correlated and the fecundities are estimated independently of tree size the goodness-of-fit and the ecological meaning of dispersal models (in the seedling or sapling stage) may be enhanced. Parameters of the UF model are estimated through the EM algorithm and their standard errors are approximated via the observed information matrix. We fit the UF model to a dataset from an expanding European beech population of central Spain as well as to a set of simulated dispersal data were the correlation between reproductive success and tree size was moderate. In comparisons with a simple allometric model, the UF model fitted the data better and the parameter estimates were less biased. We suggest using this new approach for modeling dispersal in the seedling and sapling stages when tree size (or other adult-specific covariates) is not deemed to be in strong relation to the reproductive success of adults. Models that use covariates for modeling the fecundity of adults should be preferred when reproductive success and tree size guard a strong relationship.