Biodiversity publications archive

Biodiversity and Fire: The effects and effectiveness of fire management

Proceedings of the conference held 8-9 October 1994, Footscray, Melbourne
Biodiversity Series, Paper No. 8

Biodiversity Unit
Department of the Environment, Sport and Territories, 1996

5. Long term effects of repeated burning on understorey and modelling of fire impacts

Brian Lord
School of Environmental and Information Sciences, Charles Sturt University-Murray

5.1 Abstract

A probabilistic model is offered for tracing the fate of vegetation communities in fire-prone lands that are subjected to regular fuel reduction burning. The model is based on the semi-Markov process (an extension of Markov chain modelling). The inputs necessary for the semi-Markov process are shown to be readily available, familiar to managers, or at worst, cheap and easy to procure.

By manipulation of the probabilities associated with the occurrence of low intensity prescribed fires (i.e. simulating different fire free periods), managers will be able to use readily available data to predict the long term effects of prescribed fire regimes in relation to management goals, especially where conservation is a major consideration. The model can be used to determine: mean times to episodic local extinctions of vegetation types; mean return times for naturally occurring fires; equilibrium proportions of vegetation types within a system at chosen planning horizons. In addition, by ascribing monetary (or other) values to activities associated with fuel reduction burning and fighting wildfires, it is possible to optimise the cost of entertaining the potentially conflicting goals of hazard reduction and conservation.

A case study from north-east Victoria is given to illustrate the construction of the components of the model for the fate of the understorey in open-forest, which is currently burnt on a regular rotation by the managing authority. A further case study from the mallee of central New South Wales is presented as a worked example of how the semi-Markov process could be used in decision support systems aimed at achieving conservation goals.

Key words: long-term effects, repeated burning, understorey fire impacts, probabilistic modelling.

5.2 Introduction

The last few decades in Australia have seen an acceleration of the process of identifying land that ought to be set aside in conservation reserves. To a large extent, these special places have been gazetted and put under some protection status as conservation reserves, be they national parks, biosphere reserves, nature reserves, or reference areas. Now we need to know that what we do in these reserves will conserve what's there. The legal requirements regarding minimising hazard from wildfires has seen an increased emphasis placed on fuel reduction burning that achieves the dual goals of hazard reduction and conservation.

The use of fire as a management tool in this respect has come under increased scrutiny because we have become alert to changes in the understorey caused by too many fires too soon after each other. The management response, in a lot of cases, has been to set minimum fire free periods, but there is a danger in this approach in that managers are apt to accept a fixed rotation period for fire which may, in the long run, be just as detrimental to the vegetation assemblages. The problem confronting conservation land managers is therefore one of determining the most appropriate fire regimes for a multitude of vegetation types. In this way the dual goals of hazard reduction and conservation of those vegetation types are both achieved.

Thanks to a long history of judicious forestry practice in Australia, we are generally well versed in the requirements of most of the important forest overstorey species; however, the case for the lesser plants – the forest understorey – is not so well understood. Hence, modelling the fate of understorey plant communities through time, given recurrent disturbance by fire, requires some knowledge of the responses of individual understorey species to fire and probabilities of fires returning before recovery of vulnerable species within understorey communities.

Markov models, which gained attention in ecological circles in the 1970s and 1980s, have unfortunately been largely dismissed as not accounting for major effects in pyric succession modelling (Van Hulst 1980; Usher 1981; Lippe et al. 1985); but the semi-Markov process, which is an extension of the Markov process (see Howard 1971), as developed by Moore (1989), overcomes the majority of these deficiencies. Along with advances in Markov and semi-Markov modelling came the principles of mosaic or patch dynamics (Levin 1975; Pickett & White 1985), whereby vegetation is considered as being constantly returned to early successional stages by recurrent disturbance such as fire. Complementary theories have also been advanced relating maximum diversity to intermediate rates of disturbance (Horn 1975; Huston 1979), but the quantification of "intermediate" seems elusive (see Abugov, 1982).

This paper outlines the modelling capacities of the semi-Markov process in this regard and describes the data set required as input to the model. A case study from the open-forest of north east Victoria is used to illustrate the collection of field data for input to the semi-Markov process, and an example of the workings of the semi-Markov model on informal data from the mallee of central New South Wales is given.

5.3 Fire regime and plant persistence

By having a model which incorporates expressions representing the components of the fire regime and the related effects on the persistence of plant species, we are able to manipulate the one anthropogenic input to the regime (i.e. the deliberate use of fire), in simulations of varying the fire regime.

The components of the fire regime, and the relevant effects of each component on plants, are :

  1. season – (plant phenology)
  2. frequency – (life stages/span of plant)
  3. intensity – (heat tolerance of plant tissue)
  4. areal extent – (invasion afterwards for obligate reseeders)
  5. climatic conditions following fire – (drought and plant stress, exhaustion of seed reserves etc.)

Apart from the climatic conditions component ((e) above), management (planned) fires and unplanned fires can be described in terms of these components. This paper will consider two fire classes according to this scheme:

  1. infrequent, large scale, high intensity summer fires, and
  2. frequent, small scale, low intensity spring or autumn fires.

These can be assumed to roughly equate respectively to wildfire occurrence and to fuel reduction burning programs.

5.4 Components of a probabilistic model

In its simplest form, a first order Markov process involves multiplication of a vector (representing the initial proportions of the states of a system) by a matrix (the elements of which relate to fixed probabilities of transitions between states) to achieve new sets of proportions of states at successive time steps.

Mathematically this is given by the expression :

V(t+1) = V(t).M Eq. 1


V = a vector representing the state of the system

M = a matrix of transition probabilities, and

t = time

A semi-Markov process (SMP) differs from a Markov process in that transitions need not happen at each time step, instead, a waiting time is imposed. Probabilities of transitions therefore are dependent on the initial and destination states and on the waiting times associated with the transitions (Howard 1971).

In pyric succession studies using SMP, we can model the most likely transitions using simulated alterations to the probability of the second fire type (low intensity fire) i.e. its use at certain times and under certain conditions, to effect the successional outcomes that are desired over long periods of time (planning horizons). These can also be checked against other management goals for this fire type (see Tolhurst 1995).

To model the resilience of vegetation communities to recurrent fire, the following components are therefore considered essential :

These components are all readily available or can be attained from existing or collected data with little effort.

5.4.1 Deterministic components

Specification of states

In order to trace the fate of discrete plant assemblages in a conservation sense, it is necessary to include all plant species in analysis of the vegetation suite being managed. Division of the vegetation system can then be achieved through familiar analysis techniques involving classification, ordination and/or multivariate analysis methods. These should be sufficient to get realistic groups (states) based on the factors that govern plant growth (i.e. light, temperature, water, nutrients, disturbance).

Transitions between states

Succession – some survey may be needed to obtain data from long unburnt vegetation.

Transitions caused by fire – this may require experimental fires and observation of the results of recovery (and/or literature search), but is also possible via space-for-time analysis where fire history is known or can be inferred.

The vegetation dynamics model of Noble and Slatyer (1981) which uses the functional attributes of plants, can be utilised to represent the transitions via succession and the different fire types. These transitions can be validated somewhat through studies of plant species' response to fire (e.g. Gill 1981: Benson 1985).

5.4.2 Stochastic components

Fire probabilities

Uses FDI (Fire Danger Index) from McArthur's Fire Danger Rating model (but could use Rothermel or REDBOOK model from Western Australia) along with historic weather data (see Gill 1983), and the fuel levels associated with the states as specified. Such a model can also incorporate additional factors such as standing live fuels in the shrub stratum (Cheney 1988).

Semi-Markov has an embedded Markov chain, which therefore includes all the disadvantages concerning fixed or stationary transition probabilities of the Markov process (discussed earlier). So what is needed is the incorporation of a process analogous to a non-stationary embedded Markov chain into the semi-Markov process. This can come from expressions for fuel accumulation such that the probabilities of fire in the model can increase as a function of time and fuel level increments as they do in nature.

The following parameters are therefore needed to achieve realistic expressions for fire probabilities through time:

It is the minimum fire free interval which constitutes the input variable for the model. By altering the probability of the fire class associated with fuel reduction burning (FRB), the effects of varied fire regimes can be modelled with respect to transititions between the vegetation assemblages.

Cyclic climate changes like El Nino/Southern Oscillation (ENSO) can also be incorporated by manipulating the probability of fire, particularly wildfire. The fiscal constraints could also be varied to simulate governmental (managerial) imperatives to spend more or less on FRB or other activities designed to mitigate against wildfire effects.

5.5 Existing Models

Semi-Markov modelling of the effects of fire on vegetation has been reported just twice in the literature. Henderson and Wilkins (1975) (see also Brown & Podger 1982) predicted proportions of vegetation types as a function of fire frequency for the south-west of Tasmania; and Moore (1990) gave a lucid exposition of SMP using data from Montana, USA. For this study, the SMP of Moore (1989, 1990) has been both corrected and extended to incorporate a feature analogous to a non-stationary embedded Markov process.

The correction is with respect to probabilities of fire (where more than one class of fire is used) needing to be expressed as one minus the probability of there being no fires at any time prior to the time needed for succession from the initial state to another (destination) state. Moore's (1990) claim that where two or more fire classes are used, the probability of any fire can be given simply as the sum of the probabilities of each fire type, is considered to constitute an assumption of mutually exclusive events. When the epoch length (time frame for considering transitions) is longer than one year, this can clearly not be the case, especially in Australian conditions.

Moore (1990) gives the following expression:

Gi(n) = DÂq=1 giq(n) Eq. 2.


Gi(n) =the probability of any disturbance in a community during year n of its occupancy of state i (and prior to succession).

D =the total number of fire types (q = 1...D)

giq(n) =the probability that fire type q will occur prior to succession if the vegetation is in year n of an occupancy of state i

The proposed correction is :

Gi(n) = 1- DPq=1 [1-giq(n)] Eq. 3.

which says the probability of any fire is one minus the probability of there being no fire (of either type) prior to the time for succession. The full rationale and justification for the correction will be described in a separate document.

The extension to SMP relates to the need for non-stationarity of the Markov chain embedded within the SMP. The probabilities of fire at any stage of the fuel building process for a vegetation state can be determined from the asymptotic expression of fuel accumulation, given as:

Xt = Z/k - (Z/k - X0)e-kt Eq. 4.


Xt =fuel level at time t

Z =litter fall rate

k =decomposition constant

X0 =fuel level immediately after last fire (starting fuel level)

hence, steady state (maximum) fuel level is given by Z/k

The probability of fire through time (given an ignition source), can be estimated from Xt in either of two ways :

  1. a simple fraction of the probability determined for the maximum fuel condition based on the proportion of the maximum attained, or
  2. application of the FDI and historical weather data model for that fuel level.

The second of these estimation methods is preferred but is computationally more intensive.

5.6 Inputs

The SMP therefore simply needs the states to be specified from a continuum; transitions between states to be predicted from single occurrences of disturbance by fire; the probabilities of these events occurring (and, for succession to occur, the probability of fires not occurring); and a way of checking the likelihood of any states being extinguished, either permanently or episodically.

So to summarise, the core matrix of SMP is the model. It must contain:

This then allows the calculation of the probability of each fire type for each state through time, and hence the probability of succession by default if there are no fires prior to the time needed for succession (given by (1 - Gi(n)), see Eq. 3.).

5.7 Outputs

Notwithstanding the need for correction and extension to the model, Moore (1989) has shown that the semi-Markov process can allow predictions covering :

  1. the effects of several fire types,
  2. equilibrium proportions of each state of the system,
  3. mean time to the next fire,
  4. mean time to episodic local extinctions of states,
  5. optimum management fire regimes to maximise mean times to episodic local extinctions, and
  6. the cost of optimum management fire regimes for conservation compared to the costs of fighting wildfires (and hence assistance in attaining the appropriate compromise between fuel reduction burning and conservation).

Whilst these features provide valuable information relating to conservation goals (e.g. maximising mean time to extinction of vulnerable communities), the spatial effects of recurrent disturbance such as fire in dynamic systems is difficult to incorporate (however, see Bren 1992; Lord 1993).

The core matrix for SMP constitutes a complete specification of the system - the input variable is the minimum fire free period expressed as a probability. The outputs of most interest are the proportions of each state (can be thought of as series) through time, and the mean time to episodic local extinctions of vulnerable states (vegetation communities).

By keeping the input (survey data requirements) simple and familiar (or at least easily captured), the model doesn't need anything that is either not available now, or cannot be procured with minimum cost or effort. So it will be a useful tool for managers faced with legal requirements to use fire.

5.8 Case studies

5.8.1 Case 1: Open-forest understorey

For a study of the effects of recurrent fire on the resilience of open-forest understorey communities, 80 plots have been established in north-east Victoria at Jarvis Creek Regional Park. The Park is situated 30 kms to the east of Albury-Wodonga and is primarily a red stringybark (Eucalyptus macrorhyncha) and broad leaf peppermint (E. dives) open-forest with local occurrences of spotted gum (E. mannifera ssp. mannifera), red box (E. polyanthemos), long leaf box (E. goniocalyx), narrow leaf peppermint (E. radiata), and southern blue gum (E. globulus ssp. bicostata).

Prominent understorey shrub species include silver wattle (Acacia dealbata), handsome flat-pea (Platylobium formosum), grey guinea-flower (Hibbertia obtusifolia), hop bitter-pea (Daviesia latifolia), and woolly grevillea (Grevillea lanigera), whilst the ground cover is dominated by grasses such as snowgrass
(Poa spp.), and wallaby grasses (Danthonia spp.). Bracken fern (Pteridium esculentum) is also common.

The study at Jarvis Creek has concentrated on the effects of repeated burning on the understorey but there are no satisfactory schemes for classifying understorey, so to produce discrete states and their associated transitions for analysis by SMP, multivariate analysis techniques are required. These techniques are aimed at dividing the survey plots into groups based on the species contained in them and attributing the groupings to the environmental factors associated with the plots.

Theories developed by, among others, Austin et al (1985), propose that a few crucial environmental factors are sufficient to explain the different groupings developed in this way. Modern generalised linear modelling techniques allow quantitative and qualitative (scalar) data to be used together in such analyses. So, by having relevant measurements and categories for factors that in combination control the performance of plants, meaningful states and, more importantly, transitions between states, can be determined.

The environmental factors or gradients, and the relevant parameters surveyed for the Jarvis Creek study are as follows:

Where a large proportion of the variability in the understorey composition is attributable to the disturbance gradient, postulations can be made concerning transitions between states being caused by fires. Such hypotheses obviously require testing and so a program of experimental fires, started
in 1993, is required to verify the transitions being postulated.

To establish the probabilities of two fire types (wildfire and prescribed low intensity fire), historical weather data covering 30 years will be input to a purpose-written computer program which calculates the FDI for any given fuel level and slope class. For this study, all plots were located on ground with under 10° slope.

The average number of days where various intensities of fire could be sustained can be calculated for the appropriate fuel levels. By combining data from lightning caused ignitions for the region, the probability of wildfires can be determined, and by calculating the average number of days when prescribed fires could be lit (along with management prescriptions), the probability of low intensity FRBs can also be determined.

By simulating changes to the management prescription for FRBs (to reflect policy shifts, or variations in resources available e.g. in funding or personnel), it will be possible to use SMP to determine what overall changes are likely to the nature and composition of the understorey.

5.8.2 Case 2: The mallee

Where adjoining vegetation assemblages are more discrete and the return time for fires is better known, the SMP can be utilised to predict the shifts in dominance over long planning horizons. Informal data (Bradstock, 1989a, 1989b and Bradstock,R. 1994 pers comm) on the mallee of Yathong Nature Reserve of central NSW has been used to provide a worked example of how SMP can assist in decision making concerning fire management in a conservation context. For this study only one fire type was used (wildfire), but the implications for the use of less intense fire as a management tool will become obvious.

A total of 34,000 hectares or 31per cent of Yathong Nature Reserve is covered by mallee on sand sheets and dunes. The vegetation of the dune system is typically dominated by mallee form eucalypts with an understorey of shrubs, perennial porcupine grass (Triodia irritans) and ephemeral grasses and forbs (which proliferate following good rains to make continuous fuel beds on curing). Fire sensitive mallee cypress pine (Callitris verrucosa) also occurs and can become dominant in the absence of fire. The mallee at Yathong represents the eastern limit of the current range for habitat supporting the malleefowl (Leipoa ocellata) and its significance in conservation terms is reflected in its status as a World Biosphere Reserve.

Following fire, the mallee vegetation progresses through recognisable seres which have been utilised to specify the states of the system for this study. The specified states, and two additional states representing recently burnt communities were as follows:

These are summarised in Figure 5.1.

Figure 5.1: Vegetation states selected for input to SMP from informal data of sand sheets and dunes at Yathong Nature Reserve, central NSW.

Figure 5.1: Vegetation states selected for input to SMP from informal data of sand sheets and dunes at Yathong Nature Reserve, central NSW.

Source: Bradstock 1989a, R. Bradstock year, pers. comm.

The vegetation dynamics diagram derived from the literature (e.g. Parsons 1981; Hodgkinson & Griffin 1982; Bradstock 1989a, 1989b), with attendant times for succession transitions is given as Fig. 5.2.

The probability of fire was set to reflect the reported average return time of fire (15 years) and the core matrix for SMP constructed. With this probability for fire, the outputs from SMP produced predictions on the proportions of each state which closely agreed with the extant situation for Yathong.

Calculations by SMP concerning the fire sensitive Callitris communities showed that fires occurring on average every 15 years would cause episodic local extinctions within 100 years. By manipulating the probability of fire to simulate varied fire management practices (such as judicious fire breaks; more efficient suppression efforts etc.), the mean time to episodic local extinctions could be extended to over 350 years with average return time for fire at 25 years. Beyond that time frame for fires, the mean time to episodic extinctions again decreases (see Fig. 5.3.).

Figure 5.2: State dynamics diagram for the vegetation states selected for sand sheets and dunes of the Yathong Nature Reserve.

Figure 5.2: State dynamics diagram for the vegetation states selected for sand sheets and dunes of the Yathong Nature Reserve.

Note: See Figure 5.1 for specification of the numbered states. Unbroken lines depict succession in the absence of fire (and time to succession in years), and broken lines indicate the transitions caused by fire.

Succession times were derived from Bradstock (1989a) and others.

Figure 5.3: The effect of varying fire frequency on the mean time to episodic local extinctions of communities containing mature Callitris verrucosa (mallee cypress pine) in central NSW, as modelled by the semi-Markov process.

Figure 5.3: The effect of varying fire frequency on the mean time to episodic local extinctions of communities containing mature Callitris verrucosa (mallee cypress pine) in central NSW, as modelled by the semi-Markov process.

This establishes a strong case for planned fires in vegetation suites adjacent to those containing Callitris, such that wildfires only enter some of the Callitris stands no more frequently than every 25 years.The phasing and spatial pattern of those fires remains a matter for future models, but the successful marriage of spatial and temporal models of fire effects is imminent (see Green 1987; Lord 1993), and will provide invaluable insights for land managers charged with conserving the patterns and processes that are features of Australia's fire prone landscape (see Griffin 1984; Preece 1990).

5.9 Conclusions – further research

The semi-Markov modelling technique is robust, in that it can incorporate the factors which will, in combination, dictate the successional direction of vegetation assemblages. It is therefore a potentially useful inclusion in decision support systems for planning the use of fire in fire-prone landscapes. Deficiencies inherent in constructing a semi-Markov model, particularly in relation to spatial effects, are common to any modelling methodology used to address conservation problems, but SMP offers a greater range than most. This is especially so for temporal aspects of modelling based on the essential ingredients of the dynamics of fire – fuel and weather.

Refinement of the model by more accurate specification or classification of states and transitions; greater accuracy in predicting the occurrence of fire; establishing appropriate management horizons etc., will require ongoing, long-term research. Verification of the model for any planning area can be applied following each future fire event, and validation of the model (testing its predictive ability with other systems), can only come with a genuine commitment to the collection and collation of long-term data.

Meanwhile the present acceptance of fixed rotations for fuel reduction burning, as an environmentally responsible approach to achieving joint goals of conservation and hazard reduction, must be questioned. Too little is known of the cumulative effects of the fire regimes currently imposed in vegetation suites that have been dedicated to the conservation of our natural heritage.

We all recognise that for species that rely on seeding to regenerate (obligate re-seeders) a second fire prior to seed set will force that species to local extinction. But there are few studies reporting such a loss, nor are there many that quantify the potential for such losses. For vegetation communities (i.e. special mixes of species), there would seem to be no studies at all, and yet we have legislation and public opinion that suggests we should. That a compromise must be struck between conservation goals and hazard reduction burning is now well established and models such as SMP that can simulate long term fire effects have an important part to play in achieving this compromise.

5.10 Acknowledgments

I am deeply indebted to Dr Mark Burgman (University of Melbourne) for his valued advice on classifying vegetation from data sets, to Dr Leon Bren (University of Melbourne) who first demonstrated to me the utility of Markov modelling, and to Dr Andrew Moore (CSIRO, Adelaide) who introduced me to semi-Markov modelling. For informal data on the mallee at Yathong Nature Reserve, I thank Dr Ross Bradstock.

5.11 References

Abugov, R. 1982, 'Species diversity and phasing of disturbance', Ecology, vol.63, no.2, pp.289-293.

Austin, M.P., Cunningham, R.B. & Fleming, P. 1985, 'New approaches to direct gradient analysis using environmental scalars and statistical curve-fitting procedures', in Plant Community Ecology: Papers in Honor of Robert H. Wittacker. Ed. R.K. Peet, Dr. W. Junk Publishers, Dordrecht.

Benson, D.H. 1985, 'Maturation periods for fire sensitive shrub species in Hawkesbury sandstone vegetation', Cunninghamia, vol.1, no.3,

Bradstock, R.A. 1989a, 'Dynamics of a perennial understorey', in Mediterranean Landscapes in Australia: Mallee Ecosystems and Their Management, Eds. J.C. Noble & R.A. Bradstock CSIRO, Melbourne.

Bradstock, R.A. 1989b, 'Relationships between fire regimes, plant species and fuels in mallee communities' in The Mallee Lands: A Conservation Perspective, Proceedings of the National Mallee Conference, Adelaide, April 1989. J.C. Noble, P.J. Ross, and G.K. Jones (Eds.) pp. 218-223.

Bren, L.J. 1992, 'Tree invasion of an intermittent wetland in relation to the changes in the flooding frequency of the River Murray, Australia', Australian Journal of Ecology, vol.17, no.4, pp.395-408.

Brown, M.J. & Podger, F.D. 1982, 'On the apparent anomaly between observed and predicted percentages of vegetation types in south west Tasmania', Australian Journal of Ecology, vol.7, pp.203-205.

Cheney, N.P. 1988, 'Models used for fire danger rating in Australia', in Proceedings of a Conference on Bushfire Modelling and Fire Danger Rating Systems. 11-12 July 1988, Eds. N.P. Cheney & A.M. Gill Canberra. pp.19-28.

Gill, A.M. 1981, 'Adaptive responses of Australian vascular plant species to fires', In Fire and the Australian Biota. Eds. A.M. Gill, R.H. Groves & I.R. Noble Australian Academy of Science, Canberra.

Gill, A.M. 1983, 'Causes, frequency and extent of bushfires in Victoria', in Fighting Fire ... With Fire : A Symposium on Fuel Reduction Burning in Forests. Graduate School of Environmental Science, Monash University, Melbourne, pp. 70-77.

Green, D.G. 1987, 'Spatial simulation of fire in plant communities', in Symposium. Computer Modelling and Remote Sensing in Relation to Bushfires in Australia. Canberra, June 1987. Division of National Mapping, Department of Resources and Energy, and CSIRO, Canberra.

Griffin, G.F. 1984, 'Climate, vegetation, man and fire', in: Anticipating the Inevitable: A patch-burn strategy for fire management at Uluru (Ayers Rock - Mt. Olga) National Park. ed. E.C. Saxon CSIRO. Alice Springs. pp. 7-12.

Henderson, W. & Wilkins, C.W. 1975, 'The interaction of bushfires and vegetation', Search, vol. 6., no. 4, pp. 130-133.

Hodgkinson, K.C. & Griffin, G.F. 1982, 'Adaptations of shrub species to fires in the arid zone'., in Evolution of the Flora and Fauna of Arid Australia. Eds. W.R. Barker & P.J.M. Greenslade Peacock Publications; Australian Systematic Botany Society; and ANZAAS, South Australian Div. Inc., pp. 145-152.

Horn, H.S. 1975, 'Succession', in Theoretical Ecology I: Principles and Applications. ed. R.M. May Blackwell Scientific Publications, Oxford.

Howard, R.A. 1971, 'Dynamic Probabilistic Systems Volume II: Semi-Markov and Decision Processes, John Wiley & Sons Inc., New York.

Huston, M. 1979, 'A general hypothesis of species diversity', American Naturalist, vol. 113, pp. 81-101.

Levin, S.A. 1975, 'Population dynamic models in heterogeneous environments', Annual Review of Ecological Systematics, vol. 7, pp. 287-310.

Lippe, E., de Smidt, J.P., & Glenn-Lewin, D.C. 1985, 'Markov models and succession: a test from a heathland in the Netherlands', Journal of Ecology, vol. 73, pp. 775-791.

Lord, B.L. 1993, 'Markov and semi-Markov modelling of spatial effects of fire', in The Burning Question: Fire Management in NSW. ed. J. Ross University of New England, Armidale. pp. 121-128.

Moore, A.D. 1989, 'Discrete predictive models of vegetation dynamics', PhD thesis, Australian National University.

Moore, A.D. 1990, 'The semi-Markov process: a useful tool in the analysis of vegetation dynamics for management', Journal of Environmental Management, vol. 30, pp. 111-130.

Noble, I.R. & Slatyer, R.O. 1981, 'Concepts and models of succession in vascular plant communities subject to recurrent fire', in Fire and the Australian Biota. Eds. A.M. Gill, R.H. Groves & I.R. Noble Australian Academy of Science, Canberra.

Parsons, R.F. 1981, 'Eucalyptus scrubs and shrublands', in Australian Vegetation. ed. R.H. Groves Cambridge University Press. Cambridge.
pp. 227-252.

Pickett, S.T.A. & White, P.S. 1985, 'The Ecology of Natural Disturbance and Patch Dynamics', Academic Press, Inc., Harcourt Brace Pub., Orlando, Florida.

Preece. N. 1990, 'Application of fire ecology to fire management: A manager's perspective', Proceedings of the Ecological Society of Australia, vol. 16, pp. 221-233.

Tolhurst, K. 1996, 'Effects of repeated fuel reduction burning on flora and fauna in a dry sclerophyll forest', in: Fire and Biodiversity, Biodiversity Series Paper No. 8, Biodiversity Unit, Department of the Environment, Sport and Territories, Canberra

Usher, M.D. 1981, 'Modelling ecological succession, with particular reference to Markovian models', Vegetatio, vol. 46, pp. 11-18.

Van Hulst, R. 1980, 'Vegetation dynamics or ecosystem dynamics: dynamic sufficiency in succession theory', Vegetatio, vol. 43, pp.147-151.