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ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
David G. Green
Environmental and Information Science
Charles Sturt University
PO Box 789 Albury NSW 2640 Australia
Email: dgreen@csu.edu.au
Ever since Darwin and Wallace proposed the theory of natural selection, the way in which species evolve has been a subject of intense controversy. The theory of natural selection supports the notion that species evolve by slowly accumulating changes in response to environmental pressures ("gradualism"). In recent years, however, the notion of "catastrophism" - rapid change triggered by environmental disasters - has attracted increasing attention.
The adequacy of the gradualist view to explain all aspects of evolution has been increasingly called into question in recent years. The theory of "punctuated equilibria" [3] put forward the controversial suggestion that evolutionary change does not proceed at an even rate, but in short bursts that are "punctuated" by long periods of "equilibrium". Evidence for major geologic events, such as cometary impacts [1], suggested that catastrophes may have played an important role in extinction. This view has been supported by fossil evidence for mass extinctions; (for example, [13]) and leads to the controversial hypothesis that catastophes may be responsible for all the major geologic boundaries (which are based partly on fossil evidence). However, the debate about the relationship between catastrophes and mass extinctions continues [9].
Here I argue that species evolution is an example of a general mechanism for adaptation in complex systems. This mechanism that I propose is based on properties of connectivity. It suggests that both gradualism and catastrophism play a part in evolution. Moreover, it implies that many other complex systems develop in analogous fashion.
To derive a general theory for evolution in complex systems we must first
recognise the fundamental importance of connectivity. Connectivity is
best expressed as a directed graph (X,E) ("digraph"), which is a
set X of "nodes" joined by a set
of "edges".
The universal nature of digraphs is assured by the following two theorems [7,8]. The first shows that digraphs are inherent in the structure of virtually all complex systems. The second shows that we can also regard the behaviour of complex systems as directed graphs.
An important consequence of these theorems is that properties of directed graphs underlie certain aspects of all complex systems [7,8]. Most prominent of these properties is the "connectivity avalanche" that occurs in the formation of random digraphs as edges are added to a set of nodes [4]. This avalanche effect is responsible for many kinds of phase changes in complex systems [7,8]. For example, if we represent a landscape as a grid of cells (as in cellular automata models), and represent the distribution of (say) a plant species by cells in a particular state, then we find that as the occupied proportion of the landscape increases, a phase change occurs in the size of the largest "patch" (Figure 1).
We can regard this phase change as an elementary form of chaos (a "chaotic edge"). Because of the sudden change from disconnected to connected, the system is highly sensitive to initial conditions at the phase change. Also, because of the extremely high variance the size and composition of patches in any two systems are likely to be quite different from one another.
Figure 1:
Phase changes in the connectivity of a landscape as the proportion
of "active" cells increases.
(a) The size of the largest connected "patch" of active cells.
(b) The corresponding variance in the size of the largest connected patch.
(c) Traversal time for the largest connected patch (that is, the
number of steps for an epidemic to spread throughout).
Note that the exact location of the phase change depends on
the neighbourhood function that determines connectivity.
The universality of digraphs implies that phase changes are common in the structure and behaviour of complex systems. For instance, in cellular automata [11,12] and boolean networks [10], the phase change takes the form of a transition from ordered to chaotic behaviour - the so-called "edge of chaos". Automata lying in the critical region close to this phase change in behaviour appear to possess several important properties, such as the capacity for universal computation and the ability to evolve. These observations have led to the suggestion that biological systems evolve so as to lie at or near the phase change.
Here I propose a different mechanism. It is based on observations of the structure of complex systems, rather than their behaviour. I suggest that the inherent variability of phase changes in connectivity (Figure 1(b)) provides an important source of novelty in many systems [8]. Taken in the broadest sense, we can understand variation (compare with mutation) to mean changes within a system's components or its connectivity. We can interpret selection as constraints that either prevent variation or else push it in a particular direction.
In essence I suggest that instead of adapting towards an "edge of chaos" many systems flip-flop backwards and forwards across a "chaotic edge". The mechanism (Figure 2) works as follows:
Figure 2: Biphase evolution in complex systems.
The x-axis represents a connectivity "order" parameter
appropriate to the system concerned. The spike represents
the "chaotic edge" associated with the phase change
(compare with Figure 1).
Most of the time the system sits in the phase
where selection predominates.
External stimuli force the system to flip (across the chaotic edge)
into the alternative phase where variation predominates.
The system then gradually returns, crystallising into a new
structure as it does so. See the text for further explanation.
We can see how the above mechanism applies to evolution by considering the connectivity within a landscape. Sites in a landscape are "connected" if the local populations interbreed with each other (that is, share genetic information). Dispersal is essential to maintain genetic homogeneity within populations. Should the connectivity between sites provided by dispersal fall below a critical level, then a regional population effectively breaks up into isolated sub-populations. Traditionally, isolation has been regarded as the chief method by which speciation occurs. The effect of cataclysms would be to lay waste large regions, thus isolating the surviving groups from one another and promoting speciation.
We can simulate this process (Figure 3) using a cellular automaton model to represent the "landscape" as a square grid of cells (here 400 "sites"). Randomly selected cells are occupied ("active"); other cells represent unused territory. The active cells contain a real number G to represent the phenotype of a hypothetical gene. Each cycle of the model represents a turnover of generations: random perturbations of each cell's "gene" mimic mutation; averaging with a randomly selected neighbour (if one is active) mimics sexual reproduction. An alternative "sexual" mechanism is "crossover" in which a neighbour's gene may replace the local one. Note that, here, "mutation" is simply a uniform random number r, so the "mutation rate" is simply the radius of the distribution of r (here set at 1). Tests (not shown) confirm that its only effect on the results is to set the scale for the genetic range.
Figure 3: Simulated evolution within a landscape.
The range of gene values ( G), after 10,000 "generations"
of a population that initially is: (a) homogeneous ( G = 0 everywhere);
and (b) heterogeneous (-100 < G < 100) in response to the
proportion P of active sites. Solid lines indicate sexual reproduction
by averaging; dashed lines indicate crossover.
See the text for further explanation.
Simulation (Figure 3) shows that in a fully connected landscape, reproduction acts as a spatial filter that restricts genetic drift in uniform populations (Figure 3(a)) and forces heterogeneous populations to converge (Figure 3(b)). However, if connectivity falls below the critical level, then genetic drift proceeds unimpeded in initially uniform populations (Figure 3(a)) and heterogeneous populations do not converge, but continue to drift apart (Figure 3(b)). Note that the critical region in these simulations (40-60% coverage) is specific to the neighbourhood function used and will vary according to the pattern of dispersal.
These results show that the genetic make-up of a population is highly sensitive to changes in landscape connectivity. They suggest that major landscape barriers are not necessary for speciation to occur, it suffices that overall landscape connectivity drop below the critical threshold long enough, or often enough. Conversely, they also hold a warning for conservation: populations whose distributions become restricted to small, homogeneous areas risk losing their natural genetic diversity.
If phase change triggered by disturbance is truly a common adaptive mechanism in complex systems, then other systems should provide parallels to evolution. An excellent example is provided by vegetation history. Studies, using preserved pollen, of postglacial vegetation history in Europe and northeastern North America [2] reveal that the sequences of forest changes during the last 10,000 years were remarkably uniform over vast areas. It was assumed that these "pollen zones" represent periods of more or less constant forest composition. Before the advent of radiocarbon dating, pollen zones were used to the establish relative chronologies between sites. Subsequent research has shown that these zones are associated with postglacial migrations of tree populations [2,15]. Most significantly, the zone boundaries, which are usually defined by invasions and other sudden changes in plant populations (Figure 4), often coincide with major fires [6].
Figure 4: Cataclysmic change in postglacial forests.
Pollen and charcoal records
(from Everitt Lake, Nova Scotia [6])
show that competition
from established species suppresses invaders. By clearing large areas,
major fires remove competitors and trigger explosions (see arrows) in
the size of invading tree populations.
The parallels between vegetation change and evolution are striking: pollen zones versus geologic eras, sudden changes in community composition versus mass extinctions, and major fires versus cometary impacts. This correspondence is so striking that it implies some fundamental process underlies the similarities [8]. Simulation studies imply that biotic processes in landscapes are responsible. In the case of forest change, seed dispersal acts as a conservative process [5]. Because they possess an overwhelming majority of seed sources, established species are able to outcompete invaders. By clearing large regions, major fires enable invaders to compete with established species on equal terms. Conversely, seed dispersal also enables rare species to survive in the face of superior competitors by forming clumped distributions. This process provides a mechanism for the maintenance of high diversity in tropical rainforests [5].
As the above results show, changes in connectivity play a key role in biological systems. Normally "selection" acts as a conservative process that drives systems into basins of attraction. Occasionally, disturbances alter the connectivity of the system by randomly breaking or creating connections. These changes drive the system across the critical connectivity threshold, randomly altering its structure in the process.
I suggest that the mechanism described above is an alternative to "evolution towards the edge of chaos" as described by several authors (for example, [10,12]). One notable difference is that in those studies the phase change concerned (the "edge of chaos") occurred in the system's state space; here, it is a "chaotic edge" [7,8] within the system's structure.
At this stage, it is unclear just how widespread the mechanism described here really is. It is most likely to occur where processes exist to change the structure of a complex system. For example, theoretical studies of population dynamics indicate that random assemblages usually form nonviable systems [14]; that is, the interactions form positive feedback loops, which lead to the extinction of one or more species. Now, in a system with few interactions, such instabilities are less likely than in a richly connected one. Thus gradually adding new species (for example, by migration) to a viable system increases the overall connectivity until the phase change in connectivity is reached and positive feedback becomes inevitable. The system would then lose species and collapse back to a new, subcritical state. Further immigrant species would then set the process off again.
On a small scale, biphase adaptation is extremely widespread in animal communitities. Rather than cataclysms the shifts between variation and selection in family groups are seasonal: in winter food is scarce and selection dominates; in spring and summer food is plentiful and reproduction leads to variation in the size and composition of the family group.
Finally, note that the results of the genetic simulation (Figure 3) immediately suggest an interesting approach to parallel genetic algorithms (PGA). We can exploit the parallelism to imitate the way evolution operates in a landscape. To do this we lay the population of models out across a grid of processors as if they formed a landscape. Thus crossover occurs only locally, between nearest neighbours. Instead of adjusting the mutation and crossover rates as the algorithm proceeds, we "kill" varying numbers of cells to allow others room for the offspring of surviving cells to disperse into the vacant space. Most importantly, we can allow the algorithm occasionally to kill large numbers of cells, so reducing connectivity between to below the critical level. Mortality, followed by dispersal, flips the algorithm back and forth across the connectivity threshold, so altering the balance between mutation and selection.
Evolution in Complex Systems
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