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

Ecology, June, 1999 by William A. Hoffmann

Population projections

The matrices B, A, and U were used to simulate the effect of different fire frequencies on population dynamics. It was assumed that population dynamics in the second and subsequent years following burning are similar to unburned populations. For example, to simulate population dynamics during the 4-yr period in which burning occurs in the first year, we can use

[N.sub.4] = U [multiplied by] U [multiplied by] A [multiplied by] B [multiplied by] [N.sub.0].

The vectors [N.sub.0] and [N.sub.4] are the population vectors at year 0 and year 4, respectively.

For several species, a modification of this model was necessary. For Myrsine and Miconia, it was observed that the effect of burning on reproduction continued for more than a single year; thus, additional matrices were required for the second and third years after burning. For Rourea, a matrix for the second year after burning was included.

For each fire frequency, two distinct fire regimes were simulated. The first fire regime was deterministic, so the number of years between burns remained fixed. Under the fixed fire return interval, the population vector converges onto a stable size distribution, although the size distribution depends on what stage of the fire cycle is observed. The mean annual growth rate [Lambda] is the Tth root of the dominant eigenvalue of the matrix formed by the product of the matrices used to simulate the fire regime (e.g., U [multiplied by] U [multiplied by] A [multiplied by] B, for the above example). T is the fire return interval, defined as the number of years between burns. The dominant eigenvectors and eigenvalues of the matrix models were obtained by use of a computer program that was written to simulate a fire regime until a stable size distribution is reached.

For the second fire regime, I simulated a random fire return interval by maintaining a constant probability of burning, regardless of fire history. In this case, the probability of burning in any year is I/T, where T is the average fire return interval. The mean population growth rate, [Mathematical Expression Omitted], was calculated as the geometric mean of the annual population growth rates. Each simulation was run for 6000 yr, with the first 1000 yr omitted from the calculation of [Mathematical Expression Omitted].

Elasticity, analysis

Elasticity, the proportional change in [Lambda] resulting from a proportional change in a matrix element [a.sub.ij], is defined as follows (de Kroon et al. 1986):

[e.sub.ij] [Delta] ln [Lambda]/[Delta] ln [a.sub.ij] = [a.sub.ij]/[Lambda] [multiplied by] [Delta][Lambda]/[Delta][a.sub.ij]

where [Delta][Lambda]/[Delta][a.sub.ij] is the sensitivity of [Lambda] to changes in the matrix element [a.sub.ij], holding all other matrix elements constant. The elasticities of all the matrix elements sum to one for simple matrix models (de Kroon et al. 1986) as well as periodic matrix models (Caswell and Trevisan 1994). The elasticity of a matrix element can be interpreted as the contribution of that element to the population growth rate (de Kroon et al. 1986). Elasticities were calculated for fixed fire return intervals, and were pooled to obtain sums for each of five principal transition types, shown in Fig. 1: sexual reproduction, vegetative reproduction, retrogression, stasis, and progression.