Fire and population dynamics of woody plants in a neotropical savanna: matrix model projections

Ecology, June, 1999 by William A. Hoffmann

This approximation is valid for any sufficient set of parameters (Caswell 1996a), i.e., any set of parameters which is sufficient to reconstruct the entire matrix model. The set of sufficient parameters may be simply the matrix elements; alternatively, it may be given by either of the alternative parameterizations presented by Caswell (1996a, b). Here I use another parameterization that is suited for interpreting the data at hand.

The following parameterization can be used to decompose the treatment effect on [Lambda] into the contributions due to survival, growth, and fecundity for population matrices of any form:

[Mathematical Expression Omitted] (3)

where [S.sub.i] is the probability of survival, [G.sub.ij] is the probability of growth from class j to class i, provided that the individual survives, and [F.sub.i] is fecundity. The parameterization in Expression 3 must be modified for matrices that include more than one mode of reproduction. The decomposition of the effect of burning is given by

[Mathematical Expression Omitted]. (4)

The [Delta][F.sub.i], [Delta][S.sub.i], and [Delta][G.sub.ij] are the changes in the [F.sub.i], [S.sub.i], and [G.sub.ij] due to burning. The derivatives are evaluated from the mean of the burn and unburn matrices. The contributions of the individual [G.sub.ij] will probably be of little interest. Instead, the sum of the contributions of the [G.sub.ij] for a column provides the total contribution due to changes in growth of surviving individuals.

This analysis decomposes the change in k into the contributions caused by changes in fecundity, survival, and growth under annual burning. To extend this analysis to other fire frequencies, additional modifications are necessary. For example, the difference between the population growth rate for an unburned population and the population growth rate of a population burned every three years can be decomposed as follows:

[Mathematical Expression Omitted] (5)

where [Mathematical Expression Omitted] is the change in the parameter x due to burning, relative to an unburned population, and [Mathematical Expression Omitted] is the change in the parameter in the year after burning, relative to an unburned population. The sensitivities [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are evaluated within the sequence of matrices U [multiplied by] A[prime] [multiplied by] B[prime], where the matrices A[prime] and B[prime] are defined as A[prime] = (U A)/2, and B[prime] = (U B)/2.

The decomposition of Eq. 5 can be applied to Eq. 4 to decompose the effect of burning into the contributions due to fire-induced changes in survival, growth, and fecundity for any fire frequency. For each of these classes of parameters, the contributions were summed over all size classes to obtain a single value. For each fire frequency, the contributions for the year after burning were combined with the contributions for the year of burning. These analyses were performed for fixed fire return intervals.