The model
The computer model, written in C++, describes the population dynamics of a serotinous, nonsprouting woody perennial that is subject to disturbance by fire. We chose an agebased approach to characterize the population: the model traces the number of xyearold plants, N(x), and the number of seeds stored on an xyearold plant, S(x), for each age x. Changes in N(x) and S(x) are caused by plant death, seed production, seed loss, seed release, germination and establishment of seedlings. These processes are described by the functional relationships detailed below, and illustrated conceptually in Fig. 1. The model begins with a postfire population of 100 viable seeds and calculates their survivorship and reproduction in relation to age until fire causes plant death and seed release. Fire kills all individuals, and recruitment following fire is solely from seeds.
Germination, establishment and survivorship of plants depend on the growing conditions (especially summer rainfall), and the model uses the fits eleased during the interfire period have a lower probability of germination and establishment than seeds released in response to fire (postfire release). The model takes this into account: the probability of germination and establishment for seeds released in any interfire year is only Postint of that of the postfire released seeds. For a given released seed, we estimate the probability of interfire recruitment (expressed as a proportion) to be c. 0.05 that of postfire recruitment. When a plant dies during the interfire period all seeds stored on that plant are lost. When a fire occurs all remaining plants are killed, and all of their stored seeds are released.
The mean fitness of the population (sensuStearns 1992) in relation to any set of lifehistory parameters and environmental factors is estimated using the finite rate of natural increase (λ). This is calculated as:
where n is the number of years between two fires (i.e. represents the fire interval), t is an arbitrary point in time during the simulation and N_{t} is the number of individuals at time t. Thus, N_{t} and N_{t+n} represent the number of individuals for two equivalent points (i.e. same plant age) after successive fires.
The model is deterministic in its simplest form. Mean values for life history and environmental parameters, based on information for B. hookeriana, were used as the initial settings. Parameter values and ranges used in the model are listed in Table 1. The model is generalized by examining the effects on λ of fire interval and degree of serotiny across the ranges 6–60 years and 1–1000, respectively. Changes to other parameters allowed exploration of a broad range of life histories (representing additional species) and environmental conditions (representing good and bad conditions for recruitment, survival and reproduction).
Table 1. Parameter set, initial settings (i.e. mean values for Banksia hookeriana), and range of values used in the computer model for a firekilled (nonsprouting) serotinous shrub Parameter  Notation  Initial settings  Range 


Biological attributes 
Degree of serotiny  1/r  Log_{10} (1–1000)  Step = 0.06† 
Age to first reproduction (years)  A2  5  3–15 
Age to maximum reproduction (years)  A3  15  10–25 
Longevity (years)  A4  40  30–60 
Maximum seed longevity (years)  Vm  15  2–15 
Annual rate of seed loss  Lv  0.04  0–0.20 
Viable seeds added per year at A3  Fm  200  50–1000 
Survivorship variables 
Recruitment curves*  Function  Average  Very bad–good 
Interfire recruitment  Postint  0.05  0.01–1.0 
Declining survival in old plants  D  >25  Step = 0.01 
Environmental variables 
Weather*  W  Average  Very bad–good 
Fire interval  F  6–60  Step = 1 
Standard deviation of F  sdF  0–1.0F  Step = 0.10 
Stochastic versions of the model were also analysed. A truncated, normal sampling distribution was used to select ignition times around specified mean fire intervals, giving an increasing probability of disturbance with time elapsed since last fire. Tests of the model using a Weibull distribution (Johnson & Gutsell 1994) gave equivalent results, but the truncated normal distribution handled variations in standard deviation around a constant mean fire interval in a computationally simpler manner. Simulations were repeated 1000–5000 times for each parameter set to provide a stable mean population growth surface (±SD) in relation to fire interval and degree of serotiny. The model was run for mean fire intervals from 6 to 60 years and for standard deviations around the mean fire interval ranging from 0.1 to 1.0 times the mean (i.e. coefficient of variation, CV, of 10–100%). Ignition events selected at random from the sampling distributions resulted in fires only when stand age was >5 years. This resulted in a systematic difference between the mean of the sampled distribution and the resultant mean fire interval. That is (for example), if the mean fire interval for a run was 10 years and the standard deviation was 5 (CV = 50%), then c. 68% of ignition times fell between 5 and 15 years. About 16% of ignition times, in the lower tail of the sampling distribution, were rejected (ignitions for stand ages <5 years) and new ignition times selected. The mean of the ignition times actually resulting in fires was thus greater than the mean of the theoretical distribution from which values were chosen. This difference is systematic and requires the addition of a correction factor that ranges from 4 to 6 years as fire interval increases from 6 to 20 years. Results are presented for the corrected mean fire interval.
To simulate the effects of stochastic weather, summer weather conditions for each year were sampled at random in proportion to their frequency of occurrence over the 110 years of actual rainfall records for Dongara (20:55:20:5 for good, average, bad and very bad, respectively). Based on the limited evidence for variations in inflorescence production in adult plants >15 years old, the mean rate of production was used for both average and good weather years, and 25% of the mean rate was used for bad and very bad weather years.
Sensitivity (S) of λ to changes in parameter values was calculated as:
where p represents the parameter value. For example, sensitivity for age at onset of reproduction (A2) was estimated by calculating the proportional change in λ(Δλ/λ) that resulted from a unit change in the value of A2(ΔA2/A2). The formulation of sensitivity here follows the definition of Caswell (1989).
The model is not affected by starting population size and does not limit projected plant density, although seedling survivorship rates used in the model reflect the combined effects of both densitydependent and densityindependent factors operating on the sampled populations. Nor is demographic stochasticity included. Such stochasticity is important when modelling extinction probabilities for small populations, but is unlikely to influence markedly the results of this model which explores optimum strategies for a population of assumed large size (McPeek & Kalisz 1993). Use of mean values for biological parameters also has computational advantages, making runs of the model much faster and more suitable for implementation on microcomputers.
A version of the model suitable for operation with IBMcompatible (Pentium) microcomputers, is available from the authors. The program prints the main results as a table of λvalues by levels of serotiny (51 equally spaced levels on a log_{10} scale between 1 and 1000) and fire intervals (yearly from 6 to 60 years) in a format readily imported into a variety of statistics and graphics packages.