This preprint should be cited as follows: 

Green, D.G. (1994).  Connectivity and complexity in landscapes and ecosystems. 
Pacific Conservation Biology , in press. 
------------------------------------------------------------------

Connectivity and complexity in landscapes and ecosystems

DAVID G. GREEN

Australian National University, Canberra, Australia 0200

The connectivity of sites in a landscape affects both species distribution patterns and the dynamics of whole ecosystems. Dispersal tends to produce clumped distributions, which promote species persistence and provide a possible mechanism for maintaining high species richness in tropical rainforests and other ecosystems. Simulations of multi-species systems shows that, below a critical rate, disturbance regimes have little impact on species richness. With super-critical rates of disturbance the rate of decrease in species richness depends on the balance between the rate of disturbance and dispersal range. Theoretical and simulation studies shown here reveal that landscape connectivity falls into three distinct classes: connected, disconnected, and critical. Landscape processes are inherently unpredictable when connectivity lies within the critical range. Critical levels of connectivity lead to phase changes in the behaviour of many ecosystem processes. For instance epidemics, fire spread, and invasions by exotic plants or animals are all suppressed if inter-site connectivity is too low. Conversely, genetic drift within individual populations is an order of magnitude greater if connectivity is sub-critical.

Key words:  Connectivity, Criticality, Dispersal, Evolution, Fire, Landscapes. 

INTRODUCTION

STUDIES of landscape ecology, both theoretical and field oriented, have usually focussed on the effects of environmental variations (e.g. soils) within a landscape. However, important effects of landscapes on ecosystems also arise from the ways in which different sites interact with one another. Without seed dispersal or animal migration, for instance, ecological processes such as reproduction and competition would not be possible.

Also, the effects of ecological processes across an entire landscape are often different from their effects within a small area. For instance, communities can be locally unstable, yet globally stable. Both field studies (e.g. Keller et al. 1969; van der Meijden 1979) and theoretical work (e.g. Hogeweg 1988) have show that competitive exclusion does not always occur within a landscape; competing populations can often coexist.

In the past, most models of ecosystems have ignored landscape interactions. In recent years, however, modellers have become increasingly aware of the need to incorporate spatial variation and interactions. A growing body of research, both theoretical (e.g. Wolfram 1984) and practical (e.g. on global change), shows that systems of simple interacting elements can produce extremely complex behaviour.

In this account I describe simulation results that reveal some of the ways in which landscape connectivity may influence ecosystems. I conclude by discussing some possible implications for conservation. The results presented here distill my simulation studies of landscape connectivity into three simple "experiments":

1. phase changes in landscape connectivity;
2. dispersal, disturbance and diversity; and
3. genetic drift across a landscape.

CELLULAR AUTOMATA MODELS

An increasingly popular paradigm for modelling landscapes is the "cellular automaton" (Wolfram 1984). Cellular automata (CA) models of landscapes consist of fixed arrays in which each cell represents an area of the land surface (Green 1989; Hogeweg 1988). The size of each cell fixes the scale. A cell's state corresponds to environmental features, such as coral cover or topography and neighbourhood functions simulate landscape processes (e.g. seed dispersal, epidemic spread). To represent time the models update the state of all cells in the grid iteratively, with the time step being set by the nature of the processes being simulated.

The above CA approach has many advantages. On a practical level, it is compatible both with pixel-based satellite imagery and with quadrat-based field observations. Also it enables processes that involve movement through space, e.g. dispersal, fire (Green et al. 1990), to be modelled easily. On a theoretical level, results about the behaviour of cellular automata (e.g. Wolfram 1984) can be applied to ecological systems (Green 1993). Cellular automata models have been used successfully to simulate a wide range of environmental systems, including fire spread (Green et al. 1990), starfish outbreaks (Hogeweg and Hesper 1990; Bradbury, et al. 1990), spread of disease (Pech et al. 1992; Green 1993), and forest dynamics (Green 1989).

Some of these models (e.g. Green et al. 1990; Pech et al. 1992) have been calibrated to represent real systems and have been extensively validated using field data. However the models shown below are general representations that embody, in an abstract way, features (e.g. space- filling, space-clearing) that are common to many different systems.

CONNECTIVITY AND CRITICALITY

Sites in a landscape are "connected" if there are patterns or processes to link them in some way. These links arise either from static patterns (e.g. landforms, soil distributions, contiguous forest cover) or from dynamic processes (e.g. dispersal, fire). Note that a particular landscape may have radically different degrees of connectivity with respect to different processes.

We can examine the properties of site connectivity through a simple experiment in which we treat a landscape as a vast checkerboard and consider a randomly chosen set of "active" squares (e.g. suitable growing sites for a particular plant species).

An important first question to ask about inter-site connectivity is: given a random distribution of active sites (with known density), how extensive are the regions formed by inter- connected sets of these active sites (cells)? As a simple example, suppose that each active cell is connected to its eight immediate neighbours. In this experiment, we find that the connectivity of sites in a landscape falls into three distinct phases: disconnected, critical, and connected (Fig. 1). If the density of active sites is sub-critical, then the landscape is broken into many isolated sites and small patches. If the degree is critical then a single large region may be connected (black shading in Fig. 1), but much of the landscape remains as small isolated patches. If the degree is super-critical, then almost the entire landscape is connected, with few isolated sites remaining. Note that the transition from disconnected to connected is a sharp phase change, not a continuum. That is, small shifts in connectivity can produce sudden changes in the system. In many cases the critical region itself is very narrow.

Fig. 1. Changes in connectivity of a cellular automaton as the number of "active" cells increases. The grey areas are active cells, white areas are inactive cells. The proportion of active cells increases from left to right. The black areas indicate randomly selected patches, consisting of active cells that are connected. Notice that a small change in the number of active cells produces a phase change in the system: from many small patches, isolated from one another, to essentially complete connectivity of the entire system.

The above phase changes result from an underlying directed graph ("digraph") structure (Green 1993, 1994). A digraph is simply a set of "nodes" joined by "edges". To obtain a digraph here we simply treat sites as "nodes" and links between sites (e.g. dispersal) as "edges". Landscapes thus inherit the connectivity properties of "digraphs". Perhaps the most important property is the "connectivity avalanche" (Erdos & Renyi 1960). That is, if we gradually add edges to a set of nodes, then at a certain point a sudden shift in connectivity occurs: most of the nodes suddenly join to become a single, connected "giant component", with very few nodes remaining isolated. We can see the effects of this process in landscapes by repeating the above simulation experiment to see how the size of connected regions (black areas in Fig. 1) changes with different proportions of active cells (grey shading in Fig. 1). Repeated simulations of this kind reveal the following important properties (Fig. 2):

• a small change in the number of active cells makes a large difference in the size of the largest subregion (Fig. 2a).

• the variance in the size (and composition) of the largest subregion, which is elsewhere near-zero, is maximum at the phase change (Fig. 2b);

• the "traversal time" for the largest subregion (defined to be the number of steps required for an epidemic process to cover it, starting at an arbitrary cell) is maximum (Fig. 2c). For higher coverage this time decreases as shorter routes appear.

Fig. 2. Critical changes in connectivity of a CA grid (cf Fig. 1) as the proportion of "active" cells increases (x-axes). (a) Average size of the largest connected subregion (LCS). (b) Standard deviation in the size of the LCS. (c) Traversal time for the LCS. Each point is the result of 100 iterations of a simulation in which a given proportion of cells (in a square grid of 10,000 cells) are marked as active. Note that the location of the phase change (here ~0.4) varies according to the way we define connectivity within the model grid.

The relationship of these results to criticality (Bak & Chen 1988) and percolation (Stauffer 1979; Wilkinson & Willemsen 1988) is well known. In percolation the formation of "edges" within a lattice is usually controlled by some parameter L and the phase change occurs at a critical value Lc. Perhaps the most striking examples of the above phase changes are epidemic processes:

• The spread of disease is known to depend on critical levels of infection rate and density of susceptibles and vectors (Hammersley 1966). Reducing these factors below their critical levels is a standard containment strategy (e.g. Pech et al. 1992).

• The spread of a fire across a landscape depends on the density of fuel being above a critical level; otherwise the fire quickly dies out (Green et al. 1990). This critical density is a function of environmental factors such as temperature and fuel moisture. The existence of critical conditions for fire spread has been demonstrated both experimentally (Beer 1990) and by-field observations (Gill et al. 1987).

• Studies of the spread of Crown of Thorns starfish outbreaks on the Great Barrier Reef suggest that reef to reef spread depends on inter-reef connectivity (as defined by dispersal of starfish larvae between reefs) being above a critical threshold (Bradbury et al. 1990).

• The invasion of a new species in a region depends on a critical density of sites being available for colonization. For instance pollen studies of post-glacial plant migrations in eastern Canada show that many taxa were suppressed by lack of growing sites (Green 1982, 1990).

DISPERSAL, DISTURBANCE AND DIVERSITY

The chief biotic interactions between sites in a landscape are "space-filling" processes, such as seed dispersal or animal migration, and "space-clearing" processes, such as fires, storms and other causes of mortality (Green 1989, 1990, 1992). Inter-site connectivity plays an important role in long-term changes in ecosystem composition. Short distance dispersal promotes the formation of clumped distributions (Fig. 3), which resist invasion from outsiders because of a super-abundance of local sources of seeds or other propagules (Green 1989, 1990). Clumping also promotes the persistence of species that would otherwise be eliminated by superior competitors.

Fig. 3. Spatial patterns arising in cellular automata simulations of forests ecosystems (Green, 1989). The initial distribution was random in each case. (a) "Global" dispersal - seeds spread everywhere; (b) "Local" dispersal - seeds reach adjoining cells only; (c) Local dispersal with fire; (d) As in (c), but on an environmental gradient.

Simulations of vegetation mosaics (Green 1990) imply that diversity in a landscape may depend on a critical threshold in the frequency of disturbances such as fire (Fig. 4). In the cellular automata models of vegetation shown here the "landscape" is a square grid of 2500 cells (Green 1989, 1990). The state of each cell represents the plant species that occupies the site. Each simulation (100 "years") starts with 20 identical "species", randomly distributed. Clearing elliptical patches of cells simulates the effect of severe "fires" (mild fires that do not kill the overstorey are not considered). To simulate dispersal, empty cells are "colonized" by monte carlo selection of a parent cell from a circular neighbourhood. If the average area burnt per simulation "year" is less than ~14%, then virtually all species survive. If larger areas are burnt, then the system enters a critical region in which the rate of species loss depends both on the rate of burning and on the dispersal radius.

Note that the model presented here deals only with competition between species with similar fire responses. Models of heterogeneous communities (e.g. "pioneer" versus "climax" types) show that disturbance can be necessary for the persistence of many species (Green, 1989) and hence to maintain high diversity (Connell and Slatyer 1977).

Possible consequences of the above processes include the maintenance of high diversity in tropical rainforests, formation of ecological "zones", resistance of existing communities to change and sudden, catastrophic changes in the composition of natural communities in response to clearing or fire (Green 1989, 1990). Both palaeoecological (Green 1982; Chen 1988) and simulation studies (Green 1989) indicate that short-range dispersal promotes the persistence of prevailing populations, whereas widespread fires can trigger sudden, permanent shifts in species composition. In eastern North America, for instance, the postglacial history of vegetation consisted of a series of abrupt shifts in composition (Davis 1976). The underlying causes of the shifts were changing climate and species migrations. In most cases, the shifts were triggered by major fires (Green 1982; Ritchie, 1985), which served to increase the connectivity between sites suitable for invading populations.

The tendency in the models of populations to persist because they form clumped distributions suggests a mechanism that may help to maintain high diversity in tropical rain forests. Field evidence shows that in these forests most species are small in numbers (Connell et al. 1984) and have clumped distributions (House 1985).

Fig. 4. Simulated losses of species richness within a landscape in response to different fire and dispersal regimes. "Dispersal" refers to the radius (in "cells") of the circular neighbourhood that seeds of living "plants" can reach. The "area burnt" is the average percentage of the grid that is cleared by "fires" during each simulation "year". "Species" is the number of types remaining (initally 20) after 100 simulation years. See the text for further explanation.

EVOLUTION IN A LANDSCAPE

In this experiment, if the connectivity between sites provided by dispersal falls below a critical level (here ~0.5), then a regional population effectively breaks up into isolated sub- populations. Simulation (Fig. 5) shows that in a fully connected landscape, reproduction functions as a spatial filter that restricts genetic drift in uniform populations and forces heterogeneous populations to converge. However, if connectivity falls below the critical level, then genetic drift proceeds unimpeded in initially uniform populations and heterogeneous populations do not converge, but continue to drift apart. Note that the critical region in these simulations (40-60% coverage) is specific to the neighbourhood function used and varies according to the pattern of dispersal.

Fig. 5. Simulated evolution in a landscape. The figure shows the range of gene values (G), after 10,000 "generations" of a population that initially is: (left) homogeneous (G = 0 everywhere); and (right) heterogeneous (-100 < G < 100), in response to the proportion P of active sites. Solid lines indicate sexual reproduction by averaging; dashed lines indicate crossover. In every case, the range 0.4 < P < 0.6 forms a critical region.

These results imply that the genetic makeup of a population is highly sensitive to changes in landscape connectivity. They suggest that major landscape barriers may not be not necessary for speciation to occur. Instead it suffices that overall landscape connectivity drop below the critical threshold long enough, or often enough.

DISCUSSION

Perhaps the most important lesson to result from studies of landscape connectivity is that ecosystems can be locally unstable, yet globally stable, and vice versa (Green 1989). As we saw above, short-range dispersal leads to clumped distributions that promote the persistence of populations, even in the face of a superior competitor. Conversely, fires and other disturbances can wipe out entire populations in one stroke, yet leave other populations untouched. Such results highlight the unreliability of single site models, upon which conservation strategies are often based.

Another important lesson is that landscapes processes can be inherently unpredictable. Landscapes fall into three phases: disconnected, critical, and connected. The first and last categories are highly predictable: in disconnected landscapes we can treat each element separately; in connected landscapes we can deal with the entire landscape as a single element. Where connectivity is critical, however, the size and composition of interacting elements, and hence their behaviour, are highly variable and hence unpredictable. Linked with this unpredictability is the sheer complexity of behaviour that can arise in seemingly simple systems. Although we can often specify landscape models in simple terms, their behaviour can be exceedingly complex.

The dual problems of behavioural complexity and unpredictability emphasize the need to use simulations in studying and managing landscapes. For instance, although field studies suggest many insights about landscape ecology (including some of the above) it is often impossible to test hypotheses by direct experiment. Simulation provides a means both to test hypotheses rigorously and to carry out experiments that are impractical in the field. Similarly, in landscape management we cannot expect to be able to predict the exact course of ecosystem change. Instead we need to use landscape simulations to examine possible "scenarios" and to evolve appropriate management strategies.

High landscape connectivity can have both detrimental and beneficial effects. As we have already seen, both dispersal and poor inter-site connectivity promote the persistence of populations. Poor connectivity also inhibits the spread of disturbing factors, such as fires, diseases, and introduced species. Thus we can help to minimize the threats posed by (say) diseases, by restricting contact between hosts and vectors. Fuel reduction programs not only reduce fire hazard, but also decrease the risk of fires even occurring (Gill et al. 1987). Restricting the number and area of severe wildfires is crucial for maintaining high diversity in many fire-prone systems.

Reduced landscape connectivity also leads to poor genetic communication within a population and hence to rapid genetic divergence. Major barriers, such as mountains, may not be essential for speciation to occur. The above results imply that breaking down genetic "communication" below a critical threshold could achieve the same effect. This result holds the important warning that populations restricted to small isolated pockets risk losing their genetic integrity. Genetic drift is suppressed within large populations, but populations that are restricted to small, homogeneous areas risk losing their natural genetic diversity, or of accumulating deleterious mutations that would otherwise be suppressed.

It should be noted that many of the properties discussed here for landscape ecology also apply to many other phenomena. Because the digraph structure underlying inter-site connectivity is inherent in virtually all complex systems (Green 1994), properties such as critical phase changes are shared by a wide range of systems (Green 1993). For example, in the behaviour of cellular automata and dynamic systems, changes in connectivity of the state space are responsible for a phase transition from simple to chaotic dynamics - the "edge of chaos", as described by Langton (1990) and others.

Finally, landscape connectivity plays an important role in ecosystem change. Palaeoecological studies show that changes in plant communities are often abrupt, cataclysmic events (Green 1982; Chen 1988). Simulation implies that one mechanism for sudden change is a disturbance that pushes site connectivity across a critical threshold (Green 1990). The inherent variability associated with such thresholds suggests that phase changes in connectivity may provide an important source of variety in ecosystems.

The conclusions drawn here must be qualified by the warning that the models on which they are based are extremely simple and do not account for the full complexity of real ecosystems. The kinds of interactions analyzed here, for instance, may be overlaid on a backdrop of climatic and environmental variation. Also the simple cell states used gloss over within-site factors, such as understoreys or seed banks. However, the essential message is that landscape interactions can have large effects on ecosystems. Simulations, such as those presented here, provide a useful way of examining the possible implications. They pose a challenge for field ecologists to test some of the conclusions (e.g. the existence of critical site densities) in the real world.

ACKNOWLEDGEMENTS

This work was supported by the Australian Research Council. The ANU's Connection Machine supercomputers were used for some of the simulations.

REFERENCES

• Bak, P. and Chen, K., 1988. Self-organized criticality. Phys. Rev. A 38: 364- 374.
• Beer, T. 1990. Percolation theory and fire spread. Combustion Science and Technology 72: 297-304.
• Bradbury, R.H., van der Laan, J.D. and MacDonald, B., 1990. Modelling the effects of predation and dispersal on the generation of waves of starfish outbreaks. Math. Comput. Modelling 13: 61-68.
• Chen, Y., 1988. Early Holocene population expansion of some rainforest trees at Lake Barrine basin, Queensland. Austral. J. Ecol. 13, 225-234.
• Connell, J.H., and Slatyer, R.O., 1977. Mechanisms of succession in natural communities and their role in community stability and organization. Amer. Nat. 111: 1119-1144.
• Connell, J.H., Tracey, J.G. and Webb, L.J. 1984. Compensatory recruitment growth, and mortality as factors maintaining rainforest tree diversity. Ecol. Monogr. 54: 141-164.
• Davis, M.B., 1976. Pleistocene biogeography of temperate deciduous forests. Geoscience and Man 13: 13-26.
• Erdos, P. and Renyi, A., 1960. On the evolution of random graphs. Mat. Kutato. Int. Kozl. 5: 17-61.
• Gill, A.M., Christian, K.R., Moore, P.H.R. and Forrester, R.I., 1987. Bushfire incidence, fire hazard and fuel-reduction burning. Austr. J. Ecol. 12, 299-306.
• Green, D.G., 1982. Fire and stability in the postglacial forests of southwest Nova Scotia. J. Biogeog. 9: 29-40.
• Green, D.G., 1988. Modelling forest mosaics. Pp. 584-593 in System Modelling and Optimization. ed by M.Iri & K.Yajima. Springer-Verlag: Berlin.
• Green, D.G., 1989. Simulated effects of fire, dispersal and spatial pattern on competition within vegetation mosaics. Vegetatio 82: 139-153.
• Green, D.G., 1990. Landscapes, cataclysms and population explosions. Math. Comput. Modelling 13: 75-82.
• Green, D.G., 1993. Emergent behaviour in biological systems. Pp. 25-36 in Complex Systems - From Biology to Computation. ed by D. G. Green and T. J. Bossomaier. IOS Press: Amsterdam.
• Green, D.G., 1994. Connectivity and the evolution of biological systems. J. Biol. Systems: in press.
• Green, D.G., Tridgell, A. and Gill, A.M. 1990. Interactive simulation of bushfire spread in heterogeneous fuel. Math. Comput. Modelling 13: 57-66.
• Hammersley, J.M. 1966. First passage percolation. J. R. Statist. Soc. B 28: 491-496.
• Hogeweg, P., 1988. Cellular Automata as a Paradigm for Ecological Modeling. Applied Math. Comput. 27: 81-100.
• Hogeweg, P. and Hesper, B., 1990. Crowns crowding: an individual oriented model of the Acanthaster phenomenon. Pp. 169-188 in Acanthaster and the Coral Reef: A Theoretical Perspective ed by R.H. Bradbury. Springer-Verlag: Berlin.
• House, S.M., 1985. Relationship between breeding and spatial pattern in some dioecious tropical rainforest trees. Ph.D. thesis. Austraian National University.
• Keller, B.L., Krebs, C.J. and Tamarin, R.H., 1969. Microtus population biology: demographic changes in fluctuating populations of M. ochragaster and M. pennsylvanicus in southern Indiana. Ecology 50: 587-607.
• Langton, C.G., 1990. Computation at the edge of chaos: phase transitions and emergent computation. Physica D 42: 12-37.
• Pech, R., McIlroy, J.C., Clough, M.F. and Green, D.G., 1992. A microcomputer model for predicting the spread and control of foot and mouth disease in feral pigs. Proc. 15th Vertebrate Pest Conf., Newport.
• Ritchie, J., 1985. Late Quaternary climatic and vegetation change in the lower Mackenzie Basin, northwest Canada. Ecology 66: 612-621.
• Stauffer, D., 1979. Percolation. Physics Reports 54: 1-74.
• van der Meijden, E., 1979. Herbivore exploitation of a fugitive plant species: local survival and extinction of the cinnabar moth and ragwort in a heterogeneous environment. Oecologia 42: 307-323.
• Wilkinson, D. and Willemsen, J.F., 1988. Invasion percolation: a new form of percolation theory. J. Physics A 16: 3365-3376.
• Wolfram, S., 1984. Cellular automata as models of complexity. Nature 311: 419-424.