A Population Model of Chaparral Vegetation Response to Frequent Wildfires

Abstract

The recent increase in wildfire frequency in the Santa Monica Mountains (SMM) may substantially impact plant community structure. Species of Chaparral shrubs represent the dominant vegetation type in the SMM. These species can be divided into three life history types according to their response to wildfires. Nonsprouting species are completely killed by fire and reproduce by seeds that germinate in response to a fire cue, obligate sprouting species survive by resprouting from dormant buds in a root crown because their seeds are destroyed by fire, and facultative sprouting species recover after fire both by seeds and resprouts. Based on these assumptions, we developed a set of nonlinear difference equations to model each life history type. These models can be used to predict species survivorship under varying fire return intervals. For example, frequent fires can lead to localized extinction of nonsprouting species such as Ceanothus megacarpus while several facultative sprouting species such as Ceanothus spinosus and Malosma (Rhus) laurina will persist as documented by a longitudinal study in a biological preserve in the SMM. We estimated appropriate parameter values for several chaparral species using 25 years of data and explored parameter relationships that lead to equilibrium populations. We conclude by looking at the survival strategies of these three species of chaparral shrubs under varying fire return intervals and predict changes in plant community structure under fire intervals of short return. In particular, our model predicts that an average fire return interval of greater than 12 years is required for 50 % of the initial Ceanothus megacarpus population and 25 % of the initial Ceanothus spinosus population to survive. In contrast, we predict that the Malosma laurina population will have 90 % survivorship for an average fire return interval of at least 6 years.

Appendix: Seedling and Resprout Survivorship Curves

We begin by outlining the procedure of comparing several models of seedling survivorship. The decline of seedlings in the model is based on data on Ceanothus megacarpus seedling survival from 1985–1993 that is displayed in Fig. 2. In Table 2, we compare several functions f _Y (Y _t−1,Y _t ;K _Y ,α _Y ) that can be used to model seedling decline. For each model, the parameter α _Y represents the rate of seedling decline and K _Y =gSk _Y represents the seedling survivorship, which is constant throughout a single fire-free interval. We rule out the exponential decay model because it assumes that the seedling population declines to zero and it is not a good fit of the data. Instead, we will consider several variations of the logistic and Gompertz models in the case where the initial population is greater than a sustainable population level. Besides the standard discrete logistic model described inMurray (2002), Morisita (1965) proposed a semi-implicit version equation and Hirota (1979) proposed a slightly different semi-implicit model. Morisita argues that his semi-implicit model is more natural when discretizing the continuous logistic equation. In contrast to the standard discrete logistic model, the Morisita and Hirota models have solutions that explicitly depend on time which are given in Satoh and Yamada (2002). We also consider a discrete version of the Gompertz model from Satoh (2000) and a semi-implicit version from Satoh (2003). As seen in Table 2, the standard discrete logistic model is the best fit of the Ceanothus megacarpus seedling data, but we note that all of the nonlinear models are significantly better fits than the exponential model. A visual comparison of the models from Table 2 can be found in Fig. 18. Based on the minimum residual sum of squares (RSS) and Akaike information criteria (AIC), we have chosen to use the logistic equation for nonsprouter seedling decline. We note that the Morisita equation or Hirota equation could be substituted for the logistic equation with qualitatively similar results.

Fig. 18

(Left) Visual comparison of various mathematical models to fit observed seedling survival of Ceanothus megacarpus. Models are listed in Table 2. (Right) Comparison of the logistic model to the same data

Table 2 A comparison of various models of seedling decline against the Ceanothus megacarpus data for seedling survival from 1985–1993 using a residual sum of squares of the error (RSS) and the Akaike information criterion (AIC). From Burnham and Anderson (2010), AIC=nlog(RSS/n)+2p where n is the number of data points and p is the number of parameters

We follow a similar procedure for comparing models of resprout survivorship. In Table 3 and Fig. 19, we compare several resprout decline models in a similar manner to nonsprouter seedling decline. We note that all of the models produce qualitatively similar fits. Although the exponential decay model has one less parameter and a similar AIC score to the other models, this model does not allow for a nonzero sustainable resprout population. Based on the minimum residual sum of squares and Akaike information criteria, we have chosen to use the semi-implicit Gompertz equation for resprout decline. We estimated the resprout decay α _R to be 0.1589 and the resprout survival percentage k _R to be 0.4933 which produces the fit in Fig. 19.

Fig. 19

(Left) Visual comparison of the models from Table 3 to the population declines of Ceanothus spinosus resprouts (Csr). (Right) A comparison of a simulation using the semi-implicit Gompertz model to the same data

Table 3 A comparison of various models of resprout decay against the Ceanothus spinosus data for resprout survival from 1985–1993 using a residual sum of squares of the error (RSS) and the Akaike information criterion (AIC). From Burnham and Anderson (2010), AIC=nlog(RSS/n)+2p where n is the number of data points and p is the number of parameters