A spatial model of coexistence among three Banksia species along a topographic gradient in fire-prone shrublands

Authors

  • J. Groeneveld,

    Corresponding author
    1. Department of Ecological Modelling, UFZ Centre for Environmental Research, Leipzig-Halle, Germany,
      Jürgen Groeneveld, UFZ Centre for Environmental Research Leipzig-Halle, Department of Ecological Modelling, Permoserstr. 15, D−04318 Leipzig, Germany (tel. + 49 341235 2695; fax + 49 341235 3500; e-mail groene@oesa.ufz.de).
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  • N. J. Enright,

    1. Department of Ecological Modelling, UFZ Centre for Environmental Research, Leipzig-Halle, Germany,
    2. School of Anthropology, Geography and Environmental Studies, University of Melbourne, Parkville, Australia, and
    3. Department of Environmental Biology, Curtin University of Technology, Perth, Australia
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  • B. B. Lamont,

    1. Department of Ecological Modelling, UFZ Centre for Environmental Research, Leipzig-Halle, Germany,
    2. Department of Environmental Biology, Curtin University of Technology, Perth, Australia
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  • C. Wissel

    1. Department of Ecological Modelling, UFZ Centre for Environmental Research, Leipzig-Halle, Germany,
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Jürgen Groeneveld, UFZ Centre for Environmental Research Leipzig-Halle, Department of Ecological Modelling, Permoserstr. 15, D−04318 Leipzig, Germany (tel. + 49 341235 2695; fax + 49 341235 3500; e-mail groene@oesa.ufz.de).

Summary

  • 1A spatially explicit, rule-based model for three co-occurring Banksia species was developed to investigate coexistence mediating processes in a fire-prone shrubland in western Australia. Fecundity, recruitment, mortality and other biological data for two non-sprouting (B. hookeriana, B. prionotes) and one resprouting (B. attenuata) species were available from 15 years of empirical field studies.
  • 2Without interspecific competition, each species could persist for a wide range of fire intervals (10 to > 20 years). The resprouting species performed better under shorter fire intervals (10–13 years), while both non-sprouting species were favoured by longer (15 to > 20 years) fire intervals. These results conform with those obtained from single-species, non-spatial population models.
  • 3When interspecific competition for space was included in the model, all three species exhibited optima at shorter fire intervals and with a narrower range than in isolation. The three species did not co-occur under any fire regime. At intermediate fire frequencies (11–13 years), B. hookeriana excluded the other species, while for longer intervals between fires B. prionotes became dominant.
  • 4The introduction of temporal (stochastic) variability in fire intervals (drawn from a normal distribution) failed to produce coexistence, unless spatial variability as a spatial ignition gradient was also included. The spatial arrangement of the non-sprouters observed in the field was then reproduced.
  • 5Observed patterns of coexistence and spatial distributions of all species occurred when a spatial establishment gradient for the resprouter species was included in the model (individuals of B. attenuata are known to produce more seeds in swales than on dune crests and recruit seedlings here more frequently).
  • 6Coexistence appears to be highly dependent upon the mean interfire period in combination with subtle gradients associated with fire propagation and recruitment conditions. Variation around the mean fire interval is less critical. When the system is modelled over a long time period (1500 years) coexistence is most strongly favoured for a narrow window of mean fire intervals (12–14 years).

Introduction

It is becoming increasingly apparent that the spatial arrangement of individuals has a fundamental influence on vegetation processes affecting stages such as establishment and spread. Most vegetation systems may not therefore behave in the manner described by classical non-spatial numerical analyses and models, which assume perfect mixing of individuals, so that processes act in a uniform fashion (Tilman & Kareiva 1997). Non-equilibrium versions of such models developed over the past few decades have shown that, given recurrent disturbance, temporal variation in resource availability may allow coexistence under conditions not predicted by the competitive exclusion principle (e.g. Chesson & Warner 1981; Chesson 1986; Jeltsch et al. 1998). However, several of the mechanisms thought to allow coexistence, such as the dispersal of propagules and the spread of contagious disturbances (e.g. fire), are inherently spatial but are not incorporated into such models.

Several recent studies have shown how spatial partitioning of the habitat may allow two or more species (often superior competitive and inferior ‘fugitive’ species) to coexist. For example, localized dispersal often leads to an aggregated spatial distribution and a decreased equilibrium density that may facilitate coexistence among species (Pacala 1986). The starting spatial arrangement of five annual grass species, which show a perfect inverse relationship between competitive and dispersal abilities, could greatly influence the rate of invasion of strong competitors into areas dominated by weak competitors (Silvertown et al. 1992). In a spatial model with local neighbourhood competition and random dispersal, Tilman (1994) and (Lehman & Tilman 1997) obtained stable coexistence between a potentially infinite number of competing species on a single resource. These spatial effects enhance the ability of an equal or inferior competitor to coexist in a community with an equal/superior competitor. In many spatial models, coexistence occurs as a result of trade-offs between species life-history traits; for example, inferior competitors capable of long-distance dispersal vs. superior competitors capable only of localized dispersal (Holmes & Wilson 1998).

Disturbance, at the local (e.g. individual plant death) to the landscape (e.g. fire) scales, is also inherently spatial in its character. Several modelling studies have shown that disturbance may be crucial in mediating competitive interactions and allowing coexistence (Lavorel et al. 1994; Lavorel & Chesson 1995). Interactions among the between-disturbances time-interval and species life histories are often important (Enright et al. 1998a,b). Habitat partitioning among species may occur between patches of different ages (i.e. times since disturbance), and within patches of the same age as a result of the patchy nature of many disturbances (e.g. Enright & Lamont 1989; Lamont et al. 1993). Habitat partitioning may also occur as a result of spatial heterogeneity of resource availability in the absence of disturbance, where species use different components of the available habitat, or vary in the range of habitats in which they have a competitive advantage over other species. Resource differentiation occurs when species partition the limiting resources in such a way that each is limited by a different part of the available resources (Shmida & Ellner 1984).

In Australia, fire-structured shrublands provide excellent examples of species-rich communities within which a large number of closely related species (i.e. morphologically, and presumably functionally, similar species) coexist in an environment with only subtle resource gradients. As such they provide an ideal system in which to study the mechanisms of coexistence and habitat differentiation. We investigated the circumstances facilitating the coexistence of three Banksia species (Proteaceae) in fire-prone shrublands on the Eneabba sandplain (c. 300 km north of Perth), western Australia. Two non-sprouter species (fire-killed, obligate-seeders), grow in distinct, narrowly overlapping spatial niches (B. hookeriana on the crest and slopes of low sandy dunes, and B. prionotes on the lower slopes and in the swales), while B. attenuata is a resprouter (able to regrow vegetatively after fire from buried stem tissue, and to recruit from seeds) and occurs across the whole dune-swale system. We combined the life-history attributes of the species from field data collected over 15 years to develop a spatially explicit, rule-based model. We simulated the spatial and temporal vegetation dynamics for different fire regimes and spatial gradients to understand the conditions under which the landscape scale pattern of species distributions found in the study area could be reproduced by the computer simulations (Grimm et al. 1996), and so identify the key processes mediating coexistence among these species.

Study area and species

The Eneabba sandplain receives an average rainfall of c. 500 mm year−1, of which 80% falls between May and August (winter), while maximum temperatures of > 40 °C are common between December and March. The area is characterized by unconsolidated, acidic and nutrient-poor sands, which form a low dune (up to 20 m local relief) and swale series with mean distance between consecutive dune crests of 0.5–1 km. Sands overlay clays or laterite at a depth of 60–500 cm, and are covered by a dense (cover > 70%) shrubland < 2 m tall, rich in Proteaceae (especially Banksia and Hakea species), Myrtaceae, Epacridaceae and Restionaceae, among others. Individuals of several species, including Banksia prionotes, B. menziesii, Xylomelum angustifolium (all Proteaceae) and Eucalyptus todtiana (Myrtaceae), achieve small tree size (4–8 m height) and emerge above the general shrub layer on deep sands (but are absent in areas where laterite is close to the surface). Fire is the major form of ecosystem disturbance, with an estimated natural mean return interval of about 15 years (Enright et al. 1996). Most plant species show adaptations to fire, including the ability to regrow vegetatively, soil and/or canopy (serotiny) seed storage and fire-stimulated flowering (Gill 1981). Up to eight Banksia species, most of roughly similar size, may coexist at the local scale. Important life history attributes for the three Banksia species examined here are shown in Table 1: all three are characterized by serotiny, whereby several to many years of seed production is stored in woody fruits in the plant crown, with seed release largely cued to fire, whose ignition causes the fruits to rupture.

Table 1.  Life history attributes for the three Banksia species used in the spatial model
SpeciesResponse to fireSeed productionSerotiny*Leaf morphology
  • *

    Serotiny levels are defined according to mean length of seed storage on the plant (low = 1–3 years, high = 5–10 years).

B. hookerianaNon-sprouterVery highHighNarrow/toothed
B. attenuataResprouter (from lignotuber)ModerateModerateNarrow/toothed
B. prionotesNon-sprouterHighLowBroad/toothed

The model

To investigate the long-term spatio-temporal population dynamics of Banksia hookeriana, B. prionotes and B. attenuata, a spatially explicit, individual-based model was developed. The size of the model landscape was chosen to represent typical elements of the topography of the Eneabba sandplain system. It encompasses a section of a dune-swale system 500 m in length (from dune crest to swale to crest) and 60 m in width, and is described by a grid of 200 × 24 square cells with 2.5 m sides (the approximate canopy size of a 15-year-old mature plant). Each cell is in one of five possible states: occupied by a single Banksia individual (1–3 for the three species), occupied by the regeneration phase, where one to many recruits compete for the cell for the first 5 years following fire (4), and empty (5). The following state variables were recorded in each cell: the number and species identity of individuals present; the number of seeds stored in the canopy of the mature plant occupying the cell; plant age in years; and years since the cell was last burned. At each annual time-step, a number of biological and environmental processes and events were evaluated for each grid cell. These processes and events were implemented as rules (if–then) in the model. A simplified version of the c++ program source code where the detailed implementation of all rules are given, can be found in Appendix 1. (For access to the full code, please contact the first author.)

FIRE EFFECTS

A normal distribution with specified mean and standard deviation determines the time between successive fires following the method described in Enright et al. (1998a,b). For the reference parameter set, the distribution is defined by: mean = 12, SD = 5, upper cut = 50, lower cut = 6, all values in years. The ‘cut’ values truncate the allowable sampling region under the normal curve so that fires cannot occur less than 6 or more than 50 years after the previous fire. The lower cut reflects the likely minimum time needed for a sufficient build-up of fuel to carry a fire (Enright et al. 1998a). Cutting the symmetrical normal distributions in this way shifts the mean value of the distribution to longer fire periods and decreases the standard deviation, so that the mean fire interval for the reference set is 13 ± 4.25 years. We report the dispersion of the resulting (biased) fire-interval distribution as the distance between the 5th and 95th percentiles (I90c. 13 years for the reference parameter set), rather than in terms of standard deviations.

If a grid cell is burned by a fire, non-sprouting plants are killed and all stored seeds are released. Resprouters can survive fire and regrow from a lignotuber with a certain survival probability ρfs. This survival probability depends on age (a) and the time since last fire (tf), reflecting the decline in the dormant bank of vegetative buds on the lignotuber for long fire intervals (Enright et al. 1998b):

  • 1 year ≤ a ≤ 9 years; ρfs = a/10

  • 10 years ≤ a ≤ 39 years; ρfs = 0.997

  • 40 years ≤ a ≤ 299 years; ρfs = 0.997; (for 0 < tf ≤ 20 years)

  • ρfs = (300 – a)/(300 – 40) × 0.997; (for 20 < tf ≤ 60 years)

  • ρfs = (100 – a)/(100 – 40) × 0.997; (for 60 < tf ≤ 299 years and a ≤ 100 years)

  • ρfs = 0 (for 60 < tf ≤ 299 years and a > 100 years)

The ‘plant age’ of resprouters is reduced by 2 years every time they are affected by fire, as seed production in mature individuals recommences after (the secondary juvenile phase) 2 years of post-fire vegetative regrowth. Immature individuals are similarly affected so that age at first reproduction is dependent upon the fire regime. Any canopy stored seeds on mature plants at the time of fire are dispersed (although seeds dispersed into cells already occupied by a resprouter fail to recruit).

SEED DISPERSAL

  • Seed dispersal (x) follows a negative exponential function with distance from the parent such that the probability of dispersal to distance x is

ρd(x) = exp(−x/dm)

where dm is the average dispersal distance, which varies between species relative to their potential canopy height (Table 2). The two non-sprouting species normally take 5 and 8 years (B. hookeriana and B. prionotes, respectively) to reach reproductive stage, and approximately twice these times to accumulate a canopy-stored seed bank sufficient for self-replacement after fire (Enright et al. 1996), by which time a clear ordering in height is established (Table 2). Mature individuals of the resprouter, B. attenuata, regrow rapidly after fire, so that height differences related to age should have little effect on seed dispersal distances in any of the species. For simplicity, age-height variations are not included in the modelling of dispersal.

Table 2.  Plant height (m) 15 years after fire and estimated seed dispersal distances (m) used in the model for adult individuals of the three Banksia species. See text for further details
SpeciesPlant height95% Distance*Mean distance
  • *

    Distance from plant within which 95% of seeds will fall.

B. attenuata1.5 51.67
B. hookeriana2.5 7.52.5
B. prionotes5.0155.0

Wind direction and strength records for the nearest meteorological stations (Eneabba and Dongara, western Australian Bureau of Meteorology) show no evidence of a dominant wind pattern that might influence the direction of seed dispersal. Winds alternate, on average, between easterly (land breeze) and westerly (sea breeze) from morning to afternoon in every month of the year. Thus, in the model, seeds are dispersed equally in all directions, with the angle of dispersal derived from a homogeneous distribution. We assume the periodic boundary condition, i.e. the model landscape is a torus, so that seeds that leave the grid on one border, re-enter on the other. This makes the model robust in relation to possible corridor effects, i.e. problems of limitations on plant spread along the long axis of the grid (Tilman et al. 1997). Test analyses on grids of varying widths showed that the chosen grid dimensions presented no bottleneck effects on plant dispersal (see sensitivity analysis).

POST-FIRE RECRUITMENT AND SURVIVAL

The establishment and survival of seedling cohorts for each species within cells following fire was determined from the ‘seeds required per seedling’ regression functions described in Enright et al. (1998a,b) for B. hookeriana and B. attenuata. The data used for these regression equations is derived from 15 years of field data (1983–98) in 16 sites and represent a total of 23 fires (some sites burned twice). The number of viable seeds released in the first year after each fire was estimated from cone and follicle counts and seed germination trials (Enright et al. 1996). In each succeeding year the number of seedlings recruited, and subsequently surviving to adult stage, was counted for the same plots. The number of surviving individuals at each time, t, was divided by the initial number of viable seeds released by fire for each site. These ratios were plotted against plant age (equal to time since last fire in the plots) and the regressions derived (for further details see Enright et al. 1998a). Thus, ni(t) is the number of seeds, on average, that are needed to establish an individual of age t for any age from 1 to t years. Based on recruitment and survivorship in these different sites and years, functions representing three different recruitment scenarios (poor, average and good post-fire rainfall years) are described by the model using the following equation (Enright et al. 1998a,b):

ni(rain, t) = ai(rain) × log10(t) + bi(rain)

Parameter values for ai and bi are listed in Table 3.

Table 3.  Coefficients of the seed replace functions (a × log10(age) + b). See text for details
 Bad conditions a; bAverage conditions a; bGood conditions a; b
B. hookeriana116.53; 82.1669.03; 23.5626.54; 23.03
B. prionotes102.55; 72.3060.75; 20.7623.36; 20.27
B. attenuata582.65; 410.79178.25; 120.1726.54; 23.03

We transform these seed replacement functions into the probability that a seedling of species i of age t will survive to age t+ 1 as follows:

image

These functions vary from one recruitment event to another depending on rainfall conditions, with good, average and bad years for establishment and juvenile survivorship selected at random from the probability distribution 20 : 55 : 25, using 110 years of rainfall records for the climate station (Dongara) closest to the location of field studies for these species (Enright et al. 1998a,b).

Based on empirical field data (Enright & Lamont 1992), the seedling survival rate for B. prionotes is set the same as for B. hookeriana, except in the first year when the seedling recruitment rate of B. prionotes is about 15% higher. In the first year the weather-dependent seedling recruitment rate is lower for resprouters (0.2–4% range for bad to good) than for non-sprouters (1.3–5%). With increasing age, the survival probabilities increase until a constant maximum survival probability is reached after 8 years. If there are individuals of more than one species in a cell after 5 years, then the identity of the species that will occupy the cell is determined by a lottery competition weighted by the abundance of each species (Chesson & Warner 1981; Lamont & Witkowski 1995).

The annual (interfire) mortality of mature plants (older than 8 years) is low and constant (minimum interfire mortality, Table 4) but starts, due to senescence, to increase linearly from a certain (species-dependent) threshold age (age of increased interfire mortality, Table 4).

Table 4.  Sensitivity analyses under the most complex model (scenario 5, see methods), for all parameters. Pref, Pmin and Pmax indicate the initial and ±10% values for parameters used in the model runs. S10 codes refer to sensitivities for species, and Sum is the absolute cumulative sensitivity (details of parameters are in the text and in Enright et al. 1998a,b)
ParameterPrefPminPmaxS10_hookS10_prioS10_attenSum
Mean of the fire interval distribution (years)  13  12  14 9.55 7.54−11.8428.94
Max. annual seed prod. B. hookeriana 100  90 110 8.26−5.31 −2.8316.40
Age of max. seed prod. B. hookeriana (years)  15  13  17−7.65 4.88  3.8116.34
Start of seed prod. in B. prionotes (years)   8   7   9 3.29−8.72  2.4214.43
Cell ignition probability on the crest   1   0.9   1−6.95−1.78  5.4214.15
Seedling establ. factor of B. prionotes   0.88   0.792   0.968 3.68−7.80  2.4413.92
Ignition prob. in the depression   0.7   0.63   0.77 1.27−6.51  5.9613.75
Start of seed prod. in B. hookeriana (years)   5   4   6−5.72 2.55  3.1911.46
Max. annual seed prod. B. attenuata  20  18  22−1.55−3.11  6.4311.09
Dispersal range B. attenuata (grid cells)   0.67   0.6   0.74−1.94−3.11  5.8710.93
Age at max. seed prod. B. prionotes (years)  25  22  28 2.82−6.18  1.5910.58
Lower cut of the fire period distrib. (years)   6   5   7 4.63 0.56 −3.38 8.57
Max. annual seed prod. B. prionotes 150 135 165−2.04 4.16 −1.37 7.57
I90 of fire period distribution (years)  13  12  14−4.37 0.25  2.16 6.78
Age of incr. interfire mort. B. attenuata 200 180 220−0.88−1.48  3.75 6.11
Rate of spont. fruit opening B. attenuata   0.07   0.063   0.077 0.34 1.56 −2.88 4.78
Minimum interfire mort. B. attenuata   0.0030   0.0027   0.0033 0.45 1.34 −2.50 4.29
Dispersal range B. hookeriana (grid cells)   1   0.9   1.1 1.59−1.72 −0.51 3.82
Year to max seed prod. after fire B. attenuata   7   6   8 0.97 0.89 −1.65 3.50
Rate of spont. fruit opening B. hookeriana   0.04   0.036   0.044−1.00 0.92  1.40 3.32
Width of germ. adv. zone b. attenuata (%)  46  40  52−0.93−0.83  1.46 3.21
Proportion of bad years   0.25   0.22   0.28−1.10−0.85  1.24 3.18
Rate on-plant seed loss B. hookeriana   0.04   0.036   0.044−0.94 1.61 −0.62 3.17
Minimum interfire mort. B. hookeriana   0.023   0.021   0.025−1.05 1.50  0.51 3.06
Minimum interfire mort. B. prionotes   0.015   0.014   0.017 1.23−1.29  0.45 2.97
Dispersal range B. prionotes (grid cells)   2   1.8   2.2−0.34 1.76 −0.66 2.75
Age of incr. interfire mort. B. prionotes  35  31  39−1.02 1.16 −0.53 2.71
Rate of on-plant seed loss B. attenuata   0.04   0.036   0.044 0.23 0.88 −1.56 2.67
Age at max. seed prod. B. attenuata (years)  40  36  44 0.47 0.75 −1.39 2.61
Rate of on-plant seed loss B. prionotes   0.08   0.072   0.088 0.08−1.00  1.14 2.22
Longevity B. attenuata (years) 300 270 300−0.37−0.21  1.64 2.22
Rate of spont. fruit opening B. prionotes   0.1   0.09   0.11 0.77−0.84 −0.40 2.01
Width of the grid (no. of cells)  24  22  26 0.48 0.71  0.73 1.92
Proportion of good years   0.2   0.18   0.22−0.25 0.15  1.25 1.64
Seed longevity B. prionotes (years)   5   4   6 0.30 0.81 −0.48 1.58
Age of incr. interfire mort. B. hookeriana (years)  25  22  28 0.60−0.78  0.12 1.50
Start of seed prod. B. attenuata (years)  25  22  28−0.21 0.36 −0.57 1.13
Age of incr. mort. after fire B. attenuata  40  36  44−0.45−0.43  0.11 0.99
Duration of simulation (years)150013501650−0.36−0.16  0.32 0.84
Initial time since last fire (years)  10   9  11−0.03−0.20 −0.60 0.83
Seed longevity B. attenuata (years)  10   9  11 0.01−0.27  0.51 0.79
Seed longevity B. hookeriana (years)  15  13  17−0.11 0.18 −0.38 0.67
Longevity B. hookeriana (years)  40  36  44 0.17−0.44  0.04 0.65
Upper cut of the fire period distrib. (years)  50  45  55 0.40−0.05 −0.20 0.64
Grid length (No. of cells) 200 180 220 0.26−0.12  0.07 0.45
Start of seed prod. after fire B. attenuas (years)   2   1   3 0.04−0.05 −0.17 0.26
Longevity B. prionotes (years)  60  54  66−0.05−0.04  0.14 0.23

SEED PRODUCTION AND STORAGE

The rules for seed production and storage in the canopy seed bank follow those described in the non-spatial Banksia models of Enright et al. (Enright et al. 1996; Enright et al. 1998a; Enright et al. 1998b). Seed production begins at a certain age (5, 8 and 25 years for B. hookeriana, B. prionotes and B. attenuata, respectively), and increases to a maximum seed production per year (100, 150 and 20) achieved at 15, 25 and 40 years, respectively (see Table 4). The seeds produced each year are added to the existing store of seeds. Each year, a fraction of the stored seeds is lost to on-plant granivory by insects and to decay, or through spontaneous opening of the follicles (woody fruits) in the absence of fire (see Table 4). In all three species, viable seed store on the plant reaches a stable maximum where seed production is balanced by losses (Fig. 1a). There is no evidence for a decline in seed production in old plants.

Figure 1.

(a) Estimated canopy seed store in relation to plant age for three Banksia species. (b) Estimated number of first-year seedlings per plant dispersed into neighbouring cells (excludes establishment in the grid cell of the pre-fire parent). Note: Although B. hookeriana accumulates more canopy stored seeds than B. prionotes, the latter species recruits more seedlings outside the grid cell of the mother plant, due to its greater dispersal range.

TOPOGRAPHY

When a fire occurs, the initial model assumption (homogeneous landscape) was that the whole landscape burned with equal probability (i.e. each cell is burnt with a probability of ρign = 0.8875). However, as scenarios incorporating this assumption did not allow coexistence, topography was taken into account in two ways. First, the probability of a cell burning during a fire event was made site-specific. There is evidence of moister soils (Lamont et al. 1989; Lamont & Bergl 1991), and increased patchiness of burning (B. Lamont and N. Enright, personal observation), in swales during fires in 1983 and 1991. Thus, the probability of fire is highest at the crest (all cells burned whenever there was a fire), decreasing linearly over a distance of 50–75 cells (125–187.5 m) from the crest, to a constant value of 0.7 in the swale (cells 75–125). Secondly, B. attenuata has higher recruitment rates in dune swales than on dune crests due to lower levels of summer drought stress (Lamont & Bergl 1991; Enright & Lamont 1992); its recruitment in swales was therefore modelled assuming a 60 : 40 : 0 ratio of good, average and bad years (rather than 20 : 55 : 25). The rate of change in recruitment and survival probability (from the default to the modified values) was matched linearly to the ‘topographic’ gradient that describes the change in probability of fire from crest to swale.

PARAMETER VALUES AND SENSITIVITY ANALYSES

All parameters used in the model, and the initial values for the reference parameter set, are listed in Table 4. We hypothesized that mean fire interval and the variability of the mean fire interval (described in terms of I90) are key factors controlling the landscape dynamics, and they were therefore varied across a broad range of values in order to establish the effects of the disturbance regime on coexistence and distribution pattern. All other parameters were examined across a narrower range of values, typically 10% above (phigh) and below (plow) the reference value (pref). The sensitivity of the model to changes in parameters was measured as the relative change in projected abundances (ahigh, alow and aref) of each species in the model landscape divided by the relative change in the parameter itself (phigh, plow and pref):

image

The projected abundance using the reference parameter set (aref) is the average of 1000 repetitions, all other abundances (ahigh, alow) are averaged over 100 runs. Sensitivities were calculated for each species, and as an absolute sum across all species. Sensitivities between + 1 and −1 indicate a change in abundance that is less than the proportional change in parameter value, so that we interpret sensitivity of parameters to be strong if values lie well outside this range (i.e. changes in projected abundances are greater than the relative change in parameter values).

SCENARIOS

We analysed a suite of five scenarios where each new scenario is based on its predecessor but contains one new element. In scenario 1 we used a homogeneous grid, i.e. no spatial gradient, and a constant fire period. The ignition probability was thus homogeneous in space, and the grid was randomly initialized with only one species occupying 10% of all grid cells (around 480 individuals). This scenario was run separately for each species. Scenario 2 was also based on a spatially homogeneous grid, but the grid was initialized with all three species, each occupying 10% of the grid cells, so that interspecific competition for space could occur. Scenario 3 introduces variable fire intervals drawn from the truncated normal distribution with specified variation. Scenario 4 introduces topographic variation in ignition probability and scenario 5, a gradient in recruitment conditions for B. attenuata. All simulations were run for 1500 years. At the start of each run the age of all non-sprouter individuals (i.e. of B. hookeriana and B. prionotes) was set to 10 years, while the age of the resprouter, B. attenuata, individuals was chosen at random from the range 10–299 years.

Results

SEED BANK DYNAMICS

The number of seeds stored at a given age (effectively, the time since last fire) in the crown of individuals for each of the three Banksia species is shown in Fig. 1(a). The advantage of early seed production in adult resprouters is more than offset by higher rates of seed accumulation in the non-sprouters by 9–11 years after fire. B. hookeriana stores more seeds at all ages than does B. prionotes, but the latter recruits more seedlings in contiguous cells due to greater dispersal distances, and has more seedlings that survive the first summer than does B. hookeriana (Fig. 1b).

SCENARIOS

In scenario 1 (no spatial gradient, fixed fire periods and no interspecific competition, i.e. deterministic, single species analyses) each species was able to increase in abundance for a particular range of fire periods (Fig. 2a). The resprouter, B. attenuata, increased in abundance across the landscape under relatively short fire intervals (6–20 years) with a peak in abundance around 10–12 years (when it occupied about 55% of the landscape after 1500 years of simulation). Peaks in abundance for the two non-sprouters were broader and occurred at longer intervals. Populations increased quickly once the fire interval exceeded 10 years, with B. hookeriana peaking between 12 and 20 years, and B. prionotes between 13 and 25 years (both eventually occupying > 90% of the landscape). However, only B. prionotes remained abundant at long fire intervals (> 30 years).

Figure 2.

Projected mean abundances of Banksia species (initialized with 480 individuals and maximal 4800 individuals) after 1500 years of simulation (averaged over 100 runs) for different mean values of the fire period distribution: (a) scenario 1, single species and constant fire intervals; (b) scenario 2, all species and constant fire intervals; (c) scenario 3, all species, and fire intervals drawn from a random distribution with specified I90. The scenarios all assume a spatially homogeneous landscape. Error bars are ± 1 standard deviation.

In scenario 2 (the three species present in the landscape together, and so competing for space), the range of fire periods in which each species could survive decreased to give narrow, largely non-overlapping, unimodal peaks (Fig. 2b). B. attenuata was dominant at short fire intervals (8–10 years), B. hookeriana dominated at intermediate fire intervals (11–13 years), while B. prionotes outcompeted the other species at fire intervals > 13 years. No fire regime could be identified where all three species co-occurred in the system; at least one species was always eliminated (Fig. 2b).

In scenario 3 (fire period drawn from a truncated normal distribution with specified mean and I90 values), the abundance peaks for B. attenuata (8–11 years) and B. hookeriana (11–15 years) were broader than in scenario 2, but coexistence of the three species was still not achieved (Fig. 2c). In the first two scenarios (Fig. 2a,b) the maximum abundances for each species were more or less the same, with the non-sprouting species capable of occupying more cells than the resprouting species. However, in the third scenario (Fig. 2c), peak abundances after 1500 years were slightly lower (about 40% for B. attenuata, 65% for B. hookeriana and 85% for B. prionotes), reflecting the negative impacts of occasional fires at intervals longer or shorter than the optimum for each species.

In scenario 4, we took the effect of topographic variation on the ignition probability into account. Highest coexistence probabilities were obtained at a mean fire interval of 13 years and I90 of 13 years. The averaged species abundance after 1500 years (averaged over 1000 runs) for this parametrization is shown in Fig. 3(a). Populations of all three species showed long-term persistence in the simulated landscape, and the non-sprouters established in two spatial niches: B. hookeriana on dune crests and upper slopes, and B. prionotes in swales and on lower slopes, with a small overlap in the mid-slope transition zone, as seen in the field. The resprouting species, B. attenuata, was present only on the crests. The observed coexistence pattern among all three species was not simulated unless a spatial gradient of establishment conditions for B. attenuata was included (scenario 5). While coexistence and pattern were reproduced correctly under this scenario (Fig. 3b), abundances in the transition zone (mid-slope area) were higher for B. hookeriana and lower for B. attenuata and B. prionotes than occur naturally.

Figure 3.

Relative abundances of Banksia species after 1500 years of simulation (averaged over 1000 runs) for mean fire interval = 13 years and I90 = 13 years: (a) scenario 4, all species, fire drawn from a random distribution, and an ignition gradient from dune crest to swale; (b) scenario 5, as 4 but with an establishment gradient for B. attenuata. In scenario 4 there is coexistence, but B. attenuata does not occur across the whole landscape. In scenario 5 coexistence of all three species is possible and the expected spatial pattern is reproduced. Distance on the x-axis refers to distance along the grid from dune crest (0 m) to swale (250 m) to crest (500 m).

STOCHASTICITY AND FIELD EVIDENCE

The changes in species abundances within runs differ considerably between the non-sprouting and the re-sprouting species (Fig. 4). While the numbers for the non-sprouting species fluctuate widely (Fig. 4a,b), changes in the re-sprouting species are small due to the high fire survivorship of established plants and the low levels of recruitment after most fires.

Figure 4.

Temporal dynamics of three Banksia species under a simulated fire regime with a mean fire interval of 13 years and I90 = 13 years under scenario 5: (a) B. hookeriana and B. attenuata, and (b) B. hookeriana and B. prionotes. The first 5 years after each fire (regeneration phase) are not shown. The non-sprouting and the resprouting species differ strongly in their temporal behaviour.

In order to test more precisely the ‘goodness of fit’ of the simulation results to actual landscape patterns, we compared the relative plant densities for the three Banksia species along an elevation gradient at the Mt Adams study site described in Enright & Lamont (1992) and other papers (Enright et al. 1996, 1998a,b), with results for plots derived from runs of the model under scenario 5 (Fig. 5). Because the distance between crest and swale differed for the simulated landscape and field example, the simulation plots, which were ordered along the topographic gradient from crest to swale, consisted of contiguous groups of 24 × 5 grid cells (750 m2 in area), and were compared with data for contiguous field plots of 625 m2 (25 × 25 m).

Figure 5.

Comparison of simulation model results and empirical field data for Banksia species distribution patterns along the topographic gradient from dune crest to swale (x-axis): (a) Banksia hookeriana, (b) B. prionotes, (c) B. attenuata, and (d) empirical elevation gradient and modelled fire ignition probability. Species abundances from the simulation are shown as probability bars (P at least 40 individuals ha−1) and field data as relative density among the three Banksia species. Simulation results were derived from 1000 runs evaluated after c. 1500 time steps (at 5 years after the last fire, i.e. once each cell was occupied by only one individual).

Probabilities for the presence of each species (with a minimum density of three individuals per plot, equivalent to 40 ha−1) for each location along the topographic gradient (bars in Fig. 5a–c) correspond closely with the field data, providing at least a partial validation of the model prediction of the qualitative spatial pattern. For a more quantitative comparison, more field data and parameter calibration would be necessary, but this is beyond the scope of this study.

COEXISTENCE

Given spatial heterogeneity in ignition and establishment probabilities as described in scenario 5, long-term coexistence is highly likely over only a small range of fire period distributions (Fig. 6). We calculated the fraction of all (n = 1000) runs for each fire period simulation where abundances for all three species exceeded a minimum population size (set at 5% of the maximum population size in each habitat; crest, slope, swale), to quantify a ‘coexistence probability’ for the system. With mean fire interval = 13 years and I90 = 13 years, a peak coexistence probability of 0.82 was achieved, while the extinction risk was very low (< = 0.1% after 1500 years). Coexistence was possible for mean fire intervals between 11.5 and 14 years, and variation of the distribution in the range I90 = 0–20 years (c. SD 0–7 years).

Figure 6.

Coexistence probability among the three Banksia species for different fire regimes under scenario 5 (which incorporates a spatial ignition gradient, and an establishment gradient for B. attenuata). The maximum probability of coexistence occurs at a mean fire interval of 13 years and a variability of the fire interval distribution (I90) of 13 years.

SENSITIVITY

The majority of life history and environmental parameters used in the model are robust and, when changed by ±10% from the starting value, have little effect on the average model outcomes (Table 4). The most sensitive parameters were those describing the fire period distribution (sensitivities ranked 1, 5, 7, 12, 14). Mean interfire period (fire interval) was by far the most sensitive parameter; with B. hookeriana and B. prionotes greatly advantaged by increases in the length of the interfire period, and B. attenuata greatly disadvantaged. Changes to ignition probabilities on the dune crest and in the swale were also important. Some aspects of seed bank dynamics strongly influenced the model results, especially seed production parameters for B. hookeriana and both seed production and the rate of first year seedling recruitment for B. prionotes. The abundance of all species was positively correlated with their dispersal range. This was most apparent for B. attenuata, the species with the lowest dispersal distance (although abundance in this resprouter species is also limited by low seed production). While not as important as many of the fire and life-history parameters, changes to the weather scenario (proportion of good, average and poor years for seedling recruitment) led to different outcomes for the non-sprouter and resprouter species: B. attenuata profited from increases in the proportion of years with either good or bad weather for recruitment, while B. hookeriana suffered under such changes. B. prionotes suffered when the proportion of bad years for recruitment increased, but benefited slightly when there were more good years for recruitment.

Discussion

The determinants of coexistence and habitat differentiation among three closely related Banksia species in a fire-prone shrubland were investigated using a spatially explicit, rule-based model. We first analysed species-specific responses to different fire regimes (mean fire interval and variation in fire interval) in a spatially homogeneous habitat under three simulation scenarios. Scenario 1 examined the population dynamics of each species singly, as done in previous non-spatial models for two of these Banksia species by Enright et al. (1998a,b). Our spatial model produced similar results to those of Enright et al. (1998a,b): optimum fire intervals for maximizing population size in B. attenuata and B. hookeriana were around 12–13 and 16–17 years, respectively, and the species had relatively broad adaptive peaks in relation to fire interval (8–20 and 10–25 years, respectively). In scenario 2, where all three species competed for space in the landscape the optimum fire interval for each species was shorter in every case, and the width of the adaptive peaks was narrower. This reveals the potential importance of competition for space in the population dynamic behaviour of individual species. When interspecies competition is included, the broad potential niches evident in scenario 1 (and in previous non-spatial models for these species) are reduced to much smaller realized niches (Hutchinson 1957) (see Fig. 2a,b). Perhaps more importantly, the abundance optima for the realized niches are not located at the same mean fire intervals as those for the potential niche, a result with cautionary significance for the interpretation of single species analyses.

In the deterministic case of fixed fire intervals no coexistence among the three species was possible over the period of the simulation (1500 years), although non-equilibrium disturbance models show the theoretical possibility of coexistence for competing species in a disturbance-mediated system (Huston 1979). The specific causes of the changed abundance relationships revealed in scenario 2, and failure of coexistence, appear to relate to the characteristics of the regeneration niche (Grubb 1977) on the one hand, and the fixed nature of fire intervals on the other. Lamont & Bergl (1991) obtained no evidence of niche differentiation on the basis of water relations among adults for three coexisting Banksia species (including B. hookeriana and B. attenuata) in these shrublands. Following classical equilibrium coexistence models, only one species should survive under these circumstances according to the competitive exclusion principle, as each species must have its own niche (Hardin 1960). Differences in seed production, seed dispersal in time and space, seed germination and seedling establishment (i.e. the regeneration niche, sensuGrubb 1977) have been suggested by many authors to allow violation of the exclusion principle, especially when combined with a highly temporally variable environment such as in these fire-prone shrublands (Cowling 1987; Davis 1991; Lamont & Bergl 1991).

The regeneration niches of the three species considered in the model differ from one another in several key ways. Of particular importance is the time since last fire (years) for which each species has the largest canopy store of viable seeds (see Fig. 1a). Most individuals in populations of the resprouter, Banksia attenuata, are adults that regrow vegetatively after fire and recommence seed production after only 2–3 years (in contrast, new recruits may take > 25 years to first reproduction; Enright & Lamont 1992). While it is the first among these three species to begin seed production after a fire, ultimately (> 10 years after fire) the non-sprouter species accumulate more seeds in their crown (Cowling et al. 1990) and recruit more seedlings after fire if the interval between fires is long. Recruitment of B. attenuata from seed is generally poor due to low seed supply and high mortality in the first year, especially on dune crests and slopes (Enright & Lamont 1989).

Coexistence between pairs of species (i.e. the non-sprouter B. hookeriana and the resprouter B. attenuata, or the two non-sprouters, B. hookeriana and B. prionotes) became possible (scenario 3) when temporal variability in fire intervals was introduced (contrast Fig. 2b,c). Small zones of overlap in relation to fire interval were evident, but each species still showed complete dominance of the landscape under fire regimes that maximized their abundances. Environmental variability combined with species-specific attributes can promote coexistence (Turelli & Gillespie 1980; Chesson & Warner 1981; Higgins et al. 2000), but in the present case the species-specific adaptations of plants to fire regimes were too different to allow coexistence of all three species in a homogeneous landscape.

Finally (in scenarios 4 and 5) we analysed the coexistence-promoting effects of spatial variability and habitat differentiation in the landscape. Spatial variability is an important mechanism facilitating coexistence of competing species (Chesson & Warner 1981; Tilman 1994). Previously described mathematical models have shown how spatial environmental gradients can mediate coexistence through habitat differentiation (Shigesada et al. 1979; DeAngelis & Post 1991; Pacala & Levin 1997). Over long periods of time (up to 1500 years) the spatial gradient of ignition probabilities causes such a habitat segregation and leads to the spatial separation of the two non-sprouters: B. hookeriana on crests and upper slopes, and B. prionotes on lower slopes and in swales. Areas of lower ignition probability act as refuges for the obligate seeders (Bradstock et al. 1996). For B. hookeriana, consecutive short fire intervals on dune crests and upper slopes can lead to local extinction in these parts of the landscape (Gill 1981; Zedler et al. 1983; Enright et al. 1998a). From mid-slope to lower slope, where fire is more patchy, individuals may survive at least one fire in unburned patches. By the time of the next fire they are likely to be > 20 years old and have a very large canopy seed store that assists the species to re-invade the upper slopes of the dune system when the next fire kills the plants and triggers seed release. The re-invasion from safe sites is an explicitly spatial process that depends on the spatial configuration of empty sites and the pattern of seed dispersal. The other non-sprouter, B. prionotes, is restricted to the swales and lower slopes. Its dynamics are also controlled by patchiness in fire spread, and the species is unlikely to coexist with other Banksias here unless some individuals regularly survive at least one fire in unburnt patches. It is permanently excluded from the higher parts of the landscape by frequent fire.

While temporal and spatial variability of the fire regime (scenario 4) finally led to coexistence in our model, the observed landscape scale pattern, with B. attenuata present across the whole sand dune system, could only be reproduced after the inclusion of an establishment gradient for B. attenuata (due to increased soil water availability in swales; Lamont et al. 1989; Enright & Lamont 1992). Even then, our final model result still produced a mid-slope peak in abundance for B. hookeriana that does not occur in the field. This results from the advantage that patchy fire in the lower parts of the landscape bestows upon the species in those areas (occasional old plants with many seeds help the species to saturate these areas with seeds). Under lottery competition, B. hookeriana is the most likely species to eventually dominate most cells in these areas.

Lottery (Chesson & Warner 1981), and other mathematical (Comins & Noble 1985; Warner & Chesson 1985; Yodzis 1986), models for establishment have also been proposed to explain species richness in these Banksia shrublands and other highly diverse ecosystems (Cowling 1987; Cowling 1987; Davis 1991; Lamont & Bergl 1991; Enright & Lamont 1992). Spatial and temporal variations (e.g. in time interval between fires, area and pattern of the fire, post-fire weather conditions for recruitment) make every fire a unique event, so that post-fire recruitment will differ in space and time and will not necessarily favour one species over another for long periods of time. Long-lived, but poor competitor, species like B. attenuata can survive through long periods unsuitable for recruitment due to their ability to regrow vegetatively after fire. A resprouting species is capable of maintaining a low, but stable, population, contrasting with the heavily fluctuating populations of the non-sprouters (Fig. 4). This storage effect (Warner & Chesson 1985) has been identified, in combination with other non-equilibrium model explanations (Jeltsch et al. 1998; Jeltsch et al. 2000), as the coexistence mediating process for grasses and trees in some savannahs (Higgins et al. 2000).

Seed dispersal is another crucial factor in explanations of species coexistence (Tilman 1994; Lavorel & Chesson 1995). Short mean seed dispersal distances are reported in a number of spatial models to result in patterns of intraspecific aggregation that facilitate species coexistence and community richness (Silvertown et al. 1992; Lavorel & Chesson 1995). This outcome depends to some extent at least upon the relationship between competitiveness and dispersal ability, where species with high dispersal ability can persist in sites in which superior competitors do not occur, i.e. the competitiveness/colonization ability trade-off (Tilman 1994). In contrast, the results of the sensitivity analyses (Table 4) show that increasing seed dispersal ranges always increase species abundances, so increasing their likely competitive impact (capture of cells) on other species. One reason for the advantage bestowed by increased seed dispersal ranges is the relative spatial homogeneity of the habitat for establishment (apart from the establishment gradient for B. attenuata) and the non-absorbing boundaries of the grid. This means that seeds cannot readily be lost to unsuitable habitats.

Another important factor is that seeds dispersed into the grid cell of a parent plant are likely to be wasted. This is always the situation if a resprouter survives the fire, and is also highly probable where many seeds are released into the same cell (as only one adult plant can occupy the space). Self-replacement alone is not sufficient to maintain a healthy population over a long time. The number of seeds that a plant is able to disperse outside its own cell is important in providing a mechanism for population recovery after a series of fires that may temporarily place one species at a disadvantage relative to competing species. Although B. hookeriana is capable of storing more seeds than B. prionotes, the latter disperses more seeds to contiguous cells due to its greater dispersal range. So, while B. hookeriana has a higher probability for self-replacement after a fire (in the grid cell occupied by a parent plant), B. prionotes is more likely to colonize an adjacent grid cell (so long as the fire interval is not less than 13 years). This difference in colonizing ability benefits B. prionotes in the swale, where long fire intervals are more common. However, under changed fire regimes, or in the absence of one or other species, either may occupy any part of the modelled landscape. The potential niches for each species are broadly overlapping, while the realized niches are dependent upon the detail of the disturbance regime. Changes in climate, and addition of humans as a new ignition source (first Aborigines, and more recently Europeans), can cause rapid shifts in this balance and the latter is likely to have led to reduced abundances and distribution of B. prionotes at least.

Conclusions

In most cases it is likely that coexistence will be explained by many rather than a few mechanisms (Chesson 1994). The simulation results presented here support the roles of species-specific differences in the regeneration niche, together with a highly dynamic and spatially structured environment, as an explanation for the coexistence of closely related Banksia species in the northern sandplain shrublands of south-western Australia. The role of fire regime (frequency and patchiness) was identified as a key factor in mediating coexistence, as was hypothesized based on evidence from empirical field studies (Lamont & Bergl 1991; Enright & Lamont 1992) and previous non-spatial single-species modelling (Enright et al. 1998a,b).

Our model predicts the temporal and spatial fire regimes under which coexistence is possible for a given parameterization of the demographic and seed production attributes of the three species. It provides new insights into the key factors mediating coexistence, and for conservation and management, where manipulation of fire frequency and intensity (and so patchiness) is a powerful management tool.

Acknowledgements

The data used in this paper represent a period of 15 years of field monitoring of permanent study sites. This field data collection was made possible by ARC and other research grants to NJE and BBL. Many colleagues and students assisted with the fieldwork over the years and are thanked for their help. We thank Volker Grimm and three anonymous referees for valuable comments on earlier drafts of this paper. We also appreciated the comments of the editor. Research funds from the University of Melbourne, Australia, and from the Department of Ecological Modelling, UFZ Centre for Environmental Research, Leipzig-Halle, Germany, facilitated exchange visits by NJE to Germany, and CW and JG to Australia. These were critical to the success of the modelling work.

Supplementary material

The following material is available from http://www.blackwell-science.com/products/journals/suppmat/JEC/JEC712/JEC712sm.htm:

Appendix 1 Simplified C++ pseudo code description of the spatial Banksia simulation model.

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