Mathematics and Computers in Simulation 27 (1985) 191-198 North-Holland

191

SIMULATING SPATIAL PATTERNS IN FOREST ECOSYSTEMS

D.G. GREEN, A.P.N. HOUSE and S.M. HOUSE

Department of Biogeography & Geomorphologv, Australian National University, Canberra, Australia

Simulations of spatial patterns and processes provide a means to embody hypotheses about forest ecosystems, especially when used in the interpretation of field results. The systems that we have modelled in this way include: fire spread through patchy fuels; bushfires in a forest community; breeding systems in tropical rainforest trees; and the representation of past forests and bushfires by pollen and charcoal preserved in lake sediments.

1. INTRODUCTION

Simulations are especially useful in studying forest ecosystems. The great lifespans of trees make it difficult to observe (let alone experiment with) succession and other major forest processes. While many forest simul- ations exist (e.g. 121, [15], [17]), almost all have ignored spatial pattern. In so doing they distort the representation of fire, dispersal, and other important processes. Some models have incorporated spatial pattern to a limited extent by including environ- mental gradients such as altitude or soil moisture. The few truly 2-D models that have been published have all been closely tied to experimental work (e.g. [l]).

Spatial simulations of forest communities make it possible to deduce the exact con- sequences of hypotheses about spatially extensive processes. Used in conjunction with field data, such models enable rigorous. hypothesis testing, or data interpretation. We give here four examples, all drawn from studies in progress, to illustrate how spatial simulations can supplement field experiment, or provide a substitute when experiments are not possible: (1) the spread of fire through a patchy fuel bed; (2) the impact of bushfires on species distributions and community stability; (3) the effects of the distributions of tropical rainforest trees on their breeding success; and (4) the effects of dispersal on the representation of past forests and bushfires by pollen and charcoal preserved in lake sediments.

2. MODEL STRUCTURE

'Spatial simulations' are models in which essential features (e.g. trees) are linked to specific locations. This linkage is achieved either by recording locations (usually in the form of a 2-D grid), noting what features occur at each location; or by recording a list of environmental features, noting their locations. The first method is appropriate where the environment can be subdivided into a set of discrete cells (or pixels) whose contents can be appropriately classified. At the finest scales, the cell contents repre- sent individual plants. At coarser scales, they may represent (say) patches of forest classified by their dominant vegetation. The second method of linking features to sites is more appropriate where precise locations are required and the number of objects to be examined is not too large.

As with space, time can be simulated either discretely or continuously. Discrete time steps are represented by scans through the list of objects or locations. During each scan, the list is modified for changes that have occurred during the time period supposed to have elapsed between scans. The chief danger of this approach is that non-simult- aneous events can be represented as occurring together. If these round-off errors accumu- late, so distorting (say) fire soread

patterns, then the size of the time step used must be reduced so as to separate non-simult- aneous events. Continuous time flow is simulated via Dijkstra's algorithm [5], which

037%4754/85/$3.30 0 1985, IMACS/Elsevier Science Publishers B.V. (North-Holland)

192 D.G. Green et al. / Spatial patterns in forest ecosystems

MODEL AREA BOUNDED range x= 0 100 y= 0 100 POPULATION EVOLVING size range 50 100 AGE 97 ALL trees die by this age ADULT at 19 years of age POLLEN INSECTS 20.1 m.= critical dist. SEEDING 5.5 m.= mean dispersal dist. FECUNDITY CONSTANT males 93 females 89 SURVIVAL 3 % of seeds become adults TREESIZE mean= 3 sd= 3 SPACING 3 m. min. between adult trees END

PROBLEM MODE TIME varies in this run MAPS are drawn in years 100 + 0 TIME 100 years in this run END

RUN STOP

Figure 1. Typical run control file for the rainforest simulation program SPRING. Keywords are in capitals.

orders the list of objects or locations according to the times at which specific events occur. However, in our experience this representation of time is useful only for simple epidemic phenomena (e.g. spread of a 'hot' fire).

Data structures for our models include rect- angular arrays to represent 2-D grids and linked-lists for time-ordered and population lists. Large arrays are stored in direct- access files and are processed in portions. The programming languages used are FORTRAN, PASCAL, and SIMULA. To enable flexible model structure, we use a simple keyword system to define the assumptions made, the parameter values used, and the current 'problem' to be solved (Fig. 1). All parameters receive default values unless specifically changed in the run control file. The initial forest patterns are defined either by assumptions (for hypothetical communities) or by reading in digitized maps (for runs based on field data).

3. FIRE SPREAD IN DISCRETE AND PATCHY FUEL

This model assumes that fuel is concentrated at points (e.g. grass tussocks) in a 2-D grid

[71. Fuel patchiness is represented by allowing only some (randomly located) points in the grid to contain fuel. The spread of a fire from its ignition point is modelled by computing the effect that each newly ignited point has on its neighbours. The elliptically

shaped area of scorch around a single burning point is represented by an 'ignition tem- plate'. The template is an array that gives either the heat flux at each point around the ignited point (if heat accumulation is assumed to be the chief agent of fire spread) or else the time needed for the fire to spread from the central point to other points in the array (for an epidemic 'flame contact' mechanism). Fire spread is simulated by 'laying' the ignition template over each newly ignited point (starting at the source), so determining when surrounding fuel is ignited. Increasing the size of the ignition template is equivalent to having a more nearly continuous fuel bed (dually, it can also represent a hotter fire).

Sensitivity checks [7] assessed the effects of fuel separation, wind speed (represented by the eccentricity of the elliptical ignition template), heat dissipation, fuel patchiness, and template shape. Other validation checks included comparing the model's behaviour with the results of wind tunnel tests [3] and calibrating the model against field experiments [9] to show that model-generated shapes can adequately approx- imate real fire behaviour. The calibration also confirmed that the 'heat accumulation' model for fuel ignition was valid for mild fires, while the 'flame contact' model was valid for 'hot' fires.

The model shows that fire spread is irregular (Fig. 2) when the fuel bed is both discrete (i.e. large minimum fuel separation) and patchy (i.e. low percent cover). That is, the patterns of spread are much less predictable in such fuel beds. Because of the duality between fuel separation and fire intensity, this result implies that 'hot' fires spread in more regular fashion than 'mild' fires, a conclusion that is borne out by both field experience and by wind tunnel experiments [3]. Furthermore, the model shows that fuel patchiness may be directly responsible for certain patterns of fire spread, such as the 'tear-drop' (Fig. 2). Formerly it was supposed that such shapes could form only if wind direction shifted while the fire grew.

4. BUSHFIRES AND FOREST STABILITY

These models simulate the responses of forest communities to bushfires. Cells in a 2-D grid represent 10m square quadrats (100m quadrats for planned work with satellite data) and each cell is classified according to its predominant vegetation. Noble and Slatyer

D.G. Green et al. / Spatial patterns in forest ecosystems 193

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Figure 2 : Fire shapes generated in uniform (100% cover) and patchy (25% cover) fuel beds. Crosses mark the ignition points; wind direction is down the page in each case. (a) Low wind speed, discrete fuel - note the irregular shape produced in low fuel cover; (b) High wind speed, near-continuous fuel (not to scale) - note the 'tear-drop' shape produced in patchy fuel.

[14] showed that the dynamics of fire-prone forest communities can be classified and predicted using 'vital attributes' of the dominant tree populations involved. These vital attributes include plant life stages (seedling, juvenile, adult) and stand replacement pathways for both 'normal' and fire-mediated succession.

Even the most trivial models of this sort mimic important features of forest behaviour. For example, consider a stylized communtty In which individual cells can take the values 'Bare earth', 'Grass', 'Eucalypts', or 'Rainforest'. Time varies discretely, with each scan consisting of two stages:

50 100 150 'Time'(yrs)

Figure 3 : Population instability and unpredictability resulting from randomly located ignitions. Abundances are expressed as percentages of the total area.

1.

2.

Bushfires - Ignitions occur at random locations, but fires spread only if the ignited cell contains 'grass' or 'eucalypts'. All fires grow to a fixed size and shape and leave only bare earth in the burnt area.

Forest growth - For each cell, the time T since the last fire, and vegetation V, are updated

T' =

V' =

as follows:

i 0, if fire burns out the cell T+l, otherwise

'Bare earth', if T'= 0 'Grass', if T'= 1 'Eucalypt', if T'= 5 'Rainforest', if T'= 50

V, otherwise

Crude though the model is, its behaviour closely resembles known patterns of forest dynamics:

1.

2.

Certain minimum fire frequency/area rates are needed to maintain the 'grass' and 'eucalypt' elements.

Even if the regional 'environment' is kept constant (so that the population average

194 D.G. Green et al. / Spatial patterns in forest ecosystems

remains stationary), the random location and timing of fires causes large, unpred- ictable fluctuations in population sizes (Fig. 3). Some elements (e.g. 'rainfor- est') may disappear for long periods.

3. Even though ignitions occur at random locations, the region develops 'zones' dominated by 'eucalypt' or 'rainforest' (Fig. 4).

powerful environmental force a bushfire is. Making the model more 'realistic' is simply a question of adding more features. For example, if we build environmental factors into the model, by associating appropriate variables with each location in the model 'space' and defining vegetation responses to those factors, then we find that the 'zones' mentioned above tend to coincide with dif- ferent parts of an environmental gradient.

That such simple assumptions define a system The above observations about the model's whose behaviour resembles that of systems as behaviour have profound implications when complex as whole forests emphasizes what a transformed into statements about real

Figure 4 : Fire-generated zones in a simple forest simulation. R='rainforest', e='eucalypt', t .'='grass'. See text for further explanation.

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D.G. Green et al. / Spatial patterns in forest ecosystems 195

forests. The third point, for instance, suggests that rainforests may be endangered by accumulation of unnaturally high fuel levels following disruption of presettlement fire regimes: unlike 'rainforest' in the above model, real rainforests need seed sources to regenerate. These sources (buried or newly transported seeds) would not be available if rainforest trees had been excluded from an area for long periods, as most seeds are dispersed only short distances from forest edges.

Field data from North Queensland [ll] suggest hypotheses about the role of seed transport and storage in the dynamics of rainforest/ sclerophyll boundaries. Within a relatively uniform eucalypt forest, some microsites may be 'favoured' for rainforest establishment, either by preferential dispersal of rain- forest seeds, or by greater survival rates for randomly dispersed seeds. Similarly, rainforest establishment mediated by 'predispersed' seeds stored in eucalypt soils could accelerate the fragmentation of fuel sources, given suitably long interfire periods. These potential rainforest sites may influence the distribution of ignition points and the spread of subsequent fires. Depending on fire frequency and intensity, the differ- ential flammability of rainforest and eucalypt may therefore lead to profound changes in their distributions. Data on seed fall and seed storage in both rainforest and eucalypt have been collected on a rectilinear grid of quadrats. With this information pro- viding a calibration for relevant processes, a more sophisticated version of the above model will be used to help test the above hypotheses.

While the average fire frequency for a region might be (say) 1 fire/year, some areas may burn more frequently, others not at all. Simulating the fire regime (ignition fre- quency and distribution, fuel replacement, etc) enables us to predict how the areas burnt are likely to change with increasing fire frequency (Fig. 5). Model calibration and validation will require ground data, which may be acquired by remote sensing or aerial photography (cf [HI).

by

5. BREEDING SYSTEMS IN TROPICAL RAINFOREST TREES

This model is designed to test whether different breeding systems of rainforest trees significantly affect the observed distribution patterns of their populations.

Ignitions

Figure 5 : Total area burnt versus the number of ignitions that occur before fuel is replaced. The dotted lines are predictions arising from the assumptions that (a) no fire (once started) overlaps areas previously burnt; and (b) overlap between separate fires is always proportional to the fraction of the total area already burnt. Simulation (solid line) shows that model (a) holds for small numbers of fires, and model (b) for larger numbers of fires.

Breeding mechanisms (modes of donating and receiving pollen to effect seed set) fall into two categories: (1) self-pollination and (2) transfer of pollen between trees. Self-pollinating species are capable of setting fruit in isolation, but selfed seed is often less vigorous than seed set by cross pollination between different trees. Breeding success in trees relying on cross pollination is limited by distances between trees. Spatial patterns of tree populations (Fig. 6) may be described in terms of their deviation from a uniform random distribution towards a contagious or regular distribution [4], [6]. Many rainforest species are characteris- tically common or rare; in addition both common and rare species may be locally clumped in space. Species populations which are common and/or spatially contagious typically exhibit a greater degree of cross- breeding between individuals.

196 D.G. Green et al. / Spat&d patterns rn forest ecos_gstems

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The spatial pattern exhibited by a population may therefore profoundly affect its breeding success, resulting in differential seed crops in individual trees. Since the ratio of seed set to seedling survival is normally very high, spatial configurations of potential mates in the population may be critical in determining the quantity and quality of viable offspring left by and individual. The physical spacing of successive generations is often related to the position of the parent tree, so the spatial patterns of descendants from successful and unsuccessful trees will be different. Longevity and life histories of individual trees contribute to the turn-over rate of successive generations and consequently determine the rate and direction of changes in the spatial con- figuration of different age-classes in a population. Observed patterns are the direct result of the mode of seed dispersal and differential survival of seedlings, which in turn may be influenced by the degree of breeding (pollination) success.

Critical processes simulated in the model are flower production, pollen flow, seed set, seed dispersal, seedling survival and tree longevity (Fig. 1). Detailed information for the first three of these processes and for the spatial patterns exhibited by several north Queensland rainforest tree species (Fig. 6) is being collected by one of the authors [12]. The results are being used to design and calibrate the model.

When complete the model will allow us to address several practical issues. For instance, the range of spatial configurations that will maintain stable breeding popula- tions may be predictable for some species. Potential uses in long-term forest management include predicting which species are able to survive in the face of forest disturbance (e.g. fires, selective logging) or clearance that leaves only small isolated patches of rainforest.

6. BUSHFIRES IN THE PAST

Pollen and charcoal preserved in dam, lake, and swamp sediments are rich sources of information about forests and bushfires in the past. Palynologists are now trying to interpret these records on time scales that are fine enough (single year intervals in some cases) to be relevant to field ecology

[81. Factors complicating pollen and charcoal studies at such fine resolution include variations in pollen production rates (in contrast to population changes), pollen and charcoal transport (local, water-borne components versus regional, wind-borne components) and variations in erosion and other factors affecting rates at which sediment accumulates (Fig. 7).

Some use has already been made of simulation models in palynology [17]. Simulating the processes involved in pollen and charcoal production, transport and preservation would

D.G. Green et al. / Spatial patterns in forest ecosystems 191

help to overcome the above problems of interpretation [a]. For example we can model the ways in which fires burning at different locations are likely to be represented by the charcoal records. Precise interpretations of this kind require calibration of the main parameters for a particular site (e.g. (161. Relevant field studies are currently in progress at several sites in eastern Australia.

7. CONCLUSION

Spatial processes and patterns in the environment have great influence on forest communities. An enormous amount of field research has been devoted to the problem [lo]. Such studies have suffered from a lack of ways to deduce the consequences of assumptions about the dynamic causes and effects of spatial pattern. The examples given above show that simulations can provide such a method.

Because of their visual impact (Cf. Fig. 3) spatial simulations have value as instruc- tional tools. For example, a model similar to that described in Section 4 was incorporated into a computer game to demonstrate the effects of different fire regimes. In this

game, the player acts as ranger for an imaginary national park and selects a fire control policy from the choices offered. The programme then reports on all fires that occur within the park during the next 5-10 years and on consequent vegetation changes. At the end of the period a map of the park is printed to show the player how much devastation he/she has caused.

At present the predictive power of spatial forest simulations is usually poor - forest ecosystems are too complex to permit extrac- tion of more than a few data components necessary for calibration - so their use- fulness is mostly explanatory. One possible predictive application of spatial models may

Figure 7 : Simplified model of the processes involved in pollen and charcoal preservation within a lake's catchment area. Bushfires and forests outside the catchment area contribute significantly to the air-borne components, but negligibly to the water-borne components.

PRESERVED

POLLEN

CHARCOAL

ORGANICS

INORGANIC:

198 D. G. Green et al. / Spatial patterns in forest ecosystems

lie in conjunction with remote sensing. The richness of satellite data would allow precise calibration of a model's parameters and regular monitoring of the system's performance. Perhaps the biggest problem in such an application would be to develop an adequate vegetation classification for the scales involved.

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[II

(21

[31

[41

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t71

[al

Austin, M.P., An exploratory analysis of grassland dynamics: an example of a lawn succession, Vegetatio 43 (1980) 87-94.

Botkin, D.B., Janak, J.F. and Wallis, J.R. (1972). Some consequences of a computer model of forest growth, J. Ecol. 60 (1972) 849-872.

De Mestre, N., Small-scale fire experiments (Duntroon Military College, Math. Tech. Rep. No. 3, 1981).

Diggle, P.J., On parameter estimation and goodness-of-fit testing for spatial point patterns, Biometrics 35 (1981) 87-101.

Dijkstra, E.W., A note on two problems in connection with graphs. Numer. Math. 1 (1959) 269-271.

Emmerick, J., Plant pattern analysis and radial distribution functions, Ph.D. Thesis, School of Botany, Univ. of New South Wales (November 1979).

Green, D.G., Shapes of simulated fires in discrete fuels, Ecol. Model. 20 (1983) 21-32.

Green, D.G., The ecological inter- pretation of fine resolution pollen records, New Phytol. 94 (1983) 459-477.

[g] Green, D.G., Gill, A.M. and Noble, I.R., Fire shapes and the adequacy of fire spread models, Ecol. Model. 20 (1983) 33-45.

[lo] Greig-Smith, P., Pattern in vegetation, J. Ecol. 67 (1979) 755-779.

[ll] House, A.P.N., The seed dynamics of rainforest/sclerophyll boundaries in tropical North Queensland, Ph.D. Thesis, Biogeography and Geomorphology, Austral. Nat. Univ. (in prep.).

[12] House, S.M., Breeding systems and distribution patterns in tropical rainforest trees, Ph.D. Thesis, Biogeography and Geomorphology, Austral. Nat. Univ. (in prep.).

[13] Minnich, R.A., Fire mosaics in Southern California and Northern Baja California, Science 219 (1983) 1287-1294.

[14] Noble, I.R. and Slatyer, R.O., The use of vital attributes to predict succes- sional changes in plant communities subject to recurrent disturbance, Vegetatio 43 (1980) 5-21.

[15] Shugart, H.H. and Noble, I.R., A computer model of succession and fire response of the high-altitude Eucalyptus forest of the Brindabella Range, Australian Capital Territory, Austr. J. Ecol 6 (1981) 149-164.

[16] Solomon, A.M., 1979). Pollen, in Aerobiology - the Ecological Systems Approach (Dowden, Hutchinson & Ross, Stroudsburg Penn., 1979) pp. 41-54.

[17] Solomon, A.M., Delcourt, H.R., West, D.C., and Blasing, T.J., Testing a simulation model for reconstruction of prehistoric forest-stand dynamics, Quaternary Res. 14 (1980) 275-293.