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Contrasting the edge-of-chaos and phase-shift models of evolution in complex systems. The x-axis represents a connectivity “order” parameter appropriate to the system concerned. The spike represents the critical point where a phase change occurs. (a) In the edge-of-chaos model complex systems evolve to lie near or at the critical point (the spike) between ordered and chaotic phases. (b) In the phase shift model, which is described here, external stimuli flip the system across the chaotic edge into the phase where variation predominates. The system then gradually returns, crystallizing into a new structure or behaviour as it does so. See the text for further explanation. 

Contrasting the edge-of-chaos and phase-shift models of evolution in complex systems. The x-axis represents a connectivity “order” parameter appropriate to the system concerned. The spike represents the critical point where a phase change occurs. (a) In the edge-of-chaos model complex systems evolve to lie near or at the critical point (the spike) between ordered and chaotic phases. (b) In the phase shift model, which is described here, external stimuli flip the system across the chaotic edge into the phase where variation predominates. The system then gradually returns, crystallizing into a new structure or behaviour as it does so. See the text for further explanation. 

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Here we show that connectivity and catastrophe play a key role in driving species evolution within a landscape. They also form a special case of a more general process, which occurs widely in natural and artifical systems. In this process, catastrophes cause a temporary phase change in the connectivity of a system. Different mechanisms (selection a...

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... of tree populations Davis 1976, Webb 1981). Most significantly, the zone boundaries, which are usually defined by invasions and other sudden changes in plant populations (Fig. 4), often coincide with major fires (Green 1982). Pollen and charcoal records (Green 1987) show that competition from established species suppresses invaders. By clearing large areas, major fires remove competitors and trigger explosions 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 (Green 1994b). Simulation studies imply that biotic processes in landscapes are responsible. In the case of forest change seed dispersal acts as a conservative process (Green 1989). Because they possess an overwhelming majority of seed sources, established species are able to out- compete 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 (Green 1989). One of the strongest indicators of the possibility of a universal theory of evolution is the existence of common properties underlying the structure and behaviour of all complex systems (Green 1992, 1994b, 1994c, 1999). In any complex system, connectivity is best expressed as a directed graph ( X,E ) (“digraph”). This is a set X of “nodes”, of which some or all are joined by a set E of “edges”. We represent elements of the system as nodes and interactions by edges. The universal nature of digraphs is assured by the following theorems. The first theorem shows that digraphs are inherent in all of the ways we represent complex systems. So, assuming those models are valid, digraphs are present in the structure of virtually all complex systems. The second theorem shows that we can also regard the behaviour of complex systems as directed graphs. The most important consequence of the above theorems is that properties of directed graphs explain many phenomena, such as criticality, in complex systems that had previously been treated as distinct (Green, 1993; Green, 1994b). Most prominent of these properties is the “connectivity avalanche”. Erdos and Renyi (1960) examined what happens if one takes a set of nodes and adds edges progressively to pairs of nodes chosen at random. At first the set of connected nodes are very small. But at a certain point in the procedure, a “connectivity avalanche” occurs. Adding just a few extra edges suddenly joins virtually all of the nodes into a single “giant component”. This amounts to a phase change in the system from essentially disconnected to fully connected. The above theorems show that this avalanche effect is responsible for many kinds of phase changes in complex systems (Green, 1992; Green, 1994b). For example, if we represent a landscape as a grid of cells (using the formalism of cellular automata), 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” (Fig. 2). 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. The universality of graphs in the structure and behaviour of complex systems suggests that phase changes may play a role in system evolution. In particular it enables us to generalize the model proposed above for species evolution in a landscape to identify a potentially universal mechanism based on phase changes triggered by disturbances. Here we propose that evolution is governed by a different mechanism. It is based on observations of the structure of complex systems, rather than their behaviour. We suggest that the inherent variability of phase changes in connectivity (Fig. 2b) provides an important source of novelty in many systems (Green, 1994b). Taken in the broadest sense, we can understand variation ( c.f. 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. Adopting the above idea, we suggest that many systems flip-flop backwards and forwards across a “chaotic edge” associated with a phase change in their structure or behaviour. This phase-shift mechanism (Fig. 5) works as ...

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