2002 Workshop Overview
   
 
Home Introduction Daisyworld
   
 

By Tim Lenton

There exists a long scientific tradition of searching for simple principles that govern the behaviour of complex systems. As someone working on the Gaia theory, I have been seeking such principles to help understand how the Earth system behaves. During this ongoing search, I was introduced to a body of work on the principle of maximum entropy production and its application to reconstruct the Earth's current climate state. This immediately struck a chord: Here was a simple principle that might apply to a whole class of non-equilibrium systems. It also connected to the founding idea behind the Gaia hypothesis: that Earth's atmosphere is in an extreme non-equilibrium state, and that this is in turn a product of the presence of myriad non-equilibrium life forms maintaining a highly ordered (low entropy) state. Could it be that an overarching thermodynamic principle might govern the behaviour of living organisms, of Gaia, and of many other non-equilibrium complex systems?

Having been contemplating such questions and being charged with the task of organising interesting workshops, I felt inspired to bring together researchers trying to understand and apply the principle of maximum entropy production with those working on Gaia, the Daisyworld model and other feedback control systems. The workshop took place on 4-6 September 2002 in St. Leonard's Hall at the University of Edinburgh. We began with an ascent of the Salisbury Crags and Arthur's Seat: the picturesque remnants of an extinct volcano scoured by the Quaternary ice sheets (see Figure 1). This was one of the places where James Hutton found illustrative examples for his Theory of the Earth, which marked the beginning of a systems view of our planet. The current champion of this viewpoint, Jim Lovelock (Coombe Mill, Launceston, UK) opened the formal proceedings of the workshop with a response to a recent editorial in Nature on 'Pursuing arrogant simplicities' (Campbell, 2002). In that, the Daisyworld model was described as both irritating and annoying because of its hypothetical simplicity, but given guarded praise for breaking traditional disciplinary boundaries. Jim's response was to encourage us to be as irritating as possible over the duration of the workshop!

The Daisyworld model continues to cause some irritation amongst evolutionary biologists because they see it as a special case of interaction between life and its environment that is not particularly common in the real world. In response to this, Mark Staley (Ontario, Canada) has stripped down and altered the model to make a more general version (Staley, 2002) that addresses the biologists' protests. Only one type of life - white daisies - is considered. Local variations of temperature linked to the albedo of the daisies or the bare surface are removed, but global effects on temperature are retained (this means there is only feedback on growth in the model). Growth is limited by the amount of energy (sunlight) available rather than by space, and open-ended adaptation of daisy optimum growth temperature is introduced (although some of us question this assumption). The model turns out to be an excellent regulator of temperature. In essence it assumes that a stable fixed point exists, the value of which is set by the choice of model constants. Regardless of the starting state, evolution of the growth response allows the system to move towards this equilibrium state, which is maintained by negative feedback on growth. Interestingly, when regulating in this state, the resistance of the model to temperature perturbations is at a minimum. Thus adaptation allows a homeostatic state to be reached but also makes it a more vulnerable state. There need to be constraints for regulation to occur, but it doesn't matter particularly whether these are energy limitation or space limitation (as in the original Daisyworld). Mark calls the model an example of "Adaptive Gaia": adaptation toward the moving target of an environment that is altered by the organisms. He suggested that it might be applied to the real world examples of dimethyl-sulphide (DMS) production, and biological amplification of silicate rock weathering.

It is remarkable what rich behaviour is still being discovered in variants of the simple, hypothetical Daisyworld. A recent example is the discovery of pattern formation in spatial versions of the model (Adams and Carr, submitted). Ben Adams (Heriot-Watt University, Edinburgh) had shown at our last workshop that in a 1-dimensional Daisyworld with equator-to-pole variation in incoming solar radiation, alternating stripes of white and black daisies form (wider white stripes near the equator, wider black stripes toward the pole). This is surprising because the equations of the model suggest the two daisy types should co-exist. Ben demonstrated that even on a flat surface, temperature perturbations can trigger self-organisation into a striped pattern. In two dimensions one gets an attractive pattern of concentric rings (Figure 2). Ben showed analytically how a perturbation above a certain frequency in space destabilises the co-existence solution. The behaviour is an example of diffusion-driven instability, but it differs from the better-known Turing mechanism, in that there is only one thing (temperature) diffusing, not two, and it requires a finite size of perturbation to form patterns. The pattern formation is a global phenomenon, because once daisies and temperature change in one place this affects the conditions elsewhere. As luminosity changes 'splitting' of stripes can occur. The results disrupt one assumption of the original Daisyworld: that the black and white daisies form patches of uniform temperature and unlimited size. Instead, beyond a certain size a stripe becomes unstable and the other daisy type invades. Werner von Bloh (PIK, Potsdam, Germany) showed that with 9 daisy types, stripes still occur, with a natural ordering from pale stripes near the equator to dark stripes at the pole. With stripe formation the system is assuming a low informational entropy state (low Shannon-type biodiversity). When sufficiently rapid diffusion of the daisies is introduced, the stripes disappear and a high biodiversity solution emerges with different daisy types co-existing. It was conjectured that the striped state should correspond to a high rate of entropy production due to heat transport (because of the steep temperature gradients it maintains) and that this is consistent with the formation of an ordered (low entropy) pattern.

Considering the Daisyworld studies in the context of real ecology, Dave Wilkinson (Liverpool John Moores University) summarised the current state of the Gaia debate (Lenton and Wilkinson, submitted). Recently a shift has occurred towards a view of global regulation being based predominantly on the by-products of organisms (e.g. oxygen from photosynthesis) rather than organisms' adaptively altering their local environments. This relates to a distinction in ecology between 'investment' mutualism (where both parties invest energy and/or resources) and 'by-product' mutualism (where one party benefits from the by-product of another organisms). In essence, Gaia could be 'by-product' mutualism writ large. Many scientists have difficulty grasping the concept that regulation can emerge without being selected for in a system such as the Earth. However, ecologists now accept that the regulation of population dynamics is emergent and is not due to selection at the species level. Dave noted that different groups of scientists have very different definitions of regulation: some see it as no change (i.e. perfect homeostasis with infinite gain) whereas others accept any modulation of change as regulation. Ecologists fall into the latter category, where, for example, the logistic equation for population dynamics can produce both chaos and regulation. Previous studies have shown that introducing time delays to the population equations of Daisyworld can generate chaotic variation, but it occurs about an average that is still the original regulatory temperature trajectory. The day was rounded up with a discussion of where next for Daisyworld-type modelling. The phrase "regulation for free" seemed to sum up nicely some of the recent results. It was suggested that Daisyworld could be extended to include a simple water cycle and the transfer of latent heat, and that this would help synthesize with the work being done on entropy production in the climate system.

The second day of the workshop focussed on the principle of maximum entropy production or 'MEP' for short. We were honoured to have with us Garth Paltridge (University of Tasmania, Hobart, Australia) who introduced the MEP principle in 1975 (although he expressed it then in terms of minimum entropy exchange) (Paltridge, 1975). Garth explained how in his attempts to find a simple basis for reconstructing the Earth's climate he had taken the physicist's last resort of searching for an extremum principle. He found one that allowed a remarkably accurate reconstruction of the variation of climate with latitude on Earth and a colleague noted that his formulation could readily be expressed with the dimensions of entropy. Entropy, in its original terms, is a measure of the unavailability of a system's energy to do work, and is one of the most difficult to grasp quantities in science. Garth cautioned against the use of the term by recalling an incident at a scientific meeting where a questioner was silenced by the chairperson, on asking later: "How did you know the chap was an idiot?" the chairperson's response was: "He used the word entropy in the question"!

Garth's approach was basically to consider what determines the transfer coefficient in a typical energy exchange equation, for example heat transport in a turbulent fluid such as the atmosphere. Usually the transfer coefficient is thought of as a constant (e.g. diffusion constant) or a simple function, but MEP suggests it has no simple form and can adjust to almost any value. The proposition is that it tends towards a value that corresponds to a maximum in entropy production. What has been missing is an explanation for why systems should behave in this way. Garth suggests that thinking in terms of entropy production is something of a misnomer and his recent work focuses on the dissipation that occurs in a turbulent medium. He also clarified the distinction with Prigogine's principle of minimum entropy production, which only applies to a linear system with one possible steady state. In contrast, the climate system is a non-linear system with a large number of possible steady states. Garth's recent work has focused on applying MEP at a more local scale to determine the numerous transfer coefficients in a general circulation model of Earth's climate. If the method works it will give a consistent basis for setting these parameters.

In the past, the observation that Earth's climate system is close to the MEP state was often dismissed or ignored because it was considered insufficient to demonstrate a general principle. This is the 'sample size of one' problem that also besets arguments about Gaia, for example, whether the Earth is self-regulatory by chance or whether it is a probable behaviour of a planet with abundant life. Recently, there has been a breakthrough in demonstrating the generality of MEP. Ralph Lorenz (Lunar and Planetary Laboratory, University of Arizona, USA) and his colleagues have demonstrated that Titan (Saturn's giant moon) and Mars, as well as the Earth, are all close to the maximum for entropy production due to latitudinal heat transport (Lorenz, et al., 2001). Conventional theory (which scales the latitudinal heat diffusion co-efficient with atmospheric pressure or pressure and rotation rate) under-estimates the equator-to-pole temperature contrast on Titan and over-estimates it on Mars. Assuming that the mechanisms determining the efficiency of heat transport will adjust such that entropy production is maximised leads to more accurate predictions. On each celestial body the means by which heat is transported differ somewhat, for example, on Earth water is the key condensable greenhouse gas, whereas on Mars it is carbon dioxide and on Titan it is methane.

Ralph's engineering background has led him to follow Carnot and think of systems in terms of heat engines where work is done by the heat flow from hot to cold reservoirs. The work that can be done by this heat flow depends on the difference in temperatures of the reservoirs: a larger temperature drop for a given heat flow corresponds to a higher thermodynamic efficiency, work output, and entropy production. Elaborating on this: If the heat flow is zero the entropy production is also zero, and each region of the system is in radiative equilibrium. If the heat flow is maximised the system is isothermal, and entropy production is again zero. In between, entropy production is positive and has a single maximum value. Ralph showed how one can build an electrical circuit analogue of such systems. He went on to give a fascinating overview of some of the other systems that appear to tend towards MEP (Lorenz, 2002). In the Earth's interior, assuming MEP for heat transport due to convection (relative to conduction) produces an estimate of Earth's internal thermal structure and core temperature in agreement with much more complex models. More speculatively a link may be drawn between MEP and self-organised criticality. Outstanding questions were highlighted, for example, given that heat transport between Earth's surface and the atmosphere is also close to the MEP state: What is the correct way to combine the MEP due to latitudinal and vertical heat transports? However, as Ralph highlighted, the 'big' question is why various non-equilibrium, open systems should tend towards maximising their entropy production?

We were privileged to have Roderick Dewar (INRA, Bordeaux, France) offer us a theory that can predict the behaviour of non-equilibrium systems (Dewar, submitted). This is something of a Holy Grail in physics, it being over 100 years since Boltzmann and Gibbs offered statistical expressions to describe the behaviour of equilibrium systems. In a talk aptly titled 'Climbing Mount Probable' Roderick took us through the steps of applying their logic to non-equilibrium systems. Central to this is the appreciation that the second law of thermodynamics is best explained in terms of macroscopic reproducibility. The macro-state of a system that we are most likely to observe is the one corresponding to the largest number of micro-states i.e. the most probable state (consider the classic equilibrium system example of a collection of molecules that all start in one half of a sealed container, the divider is removed and after that time they are most likely to be seen distributed throughout the container, rather than all gathered back in one half). The question is: Is there an analogue of the 2nd law in non-equilibrium systems? Roderick traced the work of Shannon in unifying the insights of Boltzmann and Gibbs and then the work of Jaynes (http://bayes.wustl.edu/etj/etj.html) in building on this to show that maximising the Gibbs-Shannon information entropy in an equilibrium or non-equilibrium system gives the most probable state (reproducible macro-state) in either case. Roderick's theory applies this logic to microscopic paths in a non-equilibrium system (rather than microscopic states in an equilibrium system), and expresses the information entropy of the system in terms of these paths. What falls out is that maximum entropy production is the most probable stationary macro-state (the one realised by the largest number of paths), subject to local conservation of energy and mass and external controls. From the theory, it is also deduced that the 2nd law holds on average, that entropy-consuming trajectories are exponentially suppressed over time (the Fluctuation Theorem), and that self-organised criticality will emerge in flux-driven systems close to the limit of zero external forcing.

Roderick has applied the new theory to leaf photosynthesis and has been able to predict observed behaviour, including the homeostasis of the ratio of carbon dioxide concentrations within the leaf versus the surrounding air (with respect to light and ambient carbon dioxide), and stomatal closure under air and soil drying. However, the observed homeostasis of leaf water potential under soil drying is not yet predicted. Such failures in the application of the theory are interesting because they tell us there exists a constraint that hasn't been taken account of. The statistical basis of MEP suggests it is a general principle, in which case it has potentially stunning implications, for example for the origin of life. Discussion focussed around the notion that homeostasis of a non-equilibrium system might simply be the maintenance of a most probable macro-state (in which, if local conservation of energy and mass holds, entropy production will tend to be maximised). The following talks focussed on further exploration and application of the MEP principle.

Hisashi Ozawa (Institute for Global Change Research, Yokohama, Japan) and his colleagues have been exploring entropy production in the climate system and numerical models of it. Hisashi opened by stressing the importance of dissipative structures (e.g. living organisms) within the Earth climate system as converters of low entropy solar photons to high entropy infrared radiation. His theoretical work has unified theories of thermal convection and shear turbulence with the MEP principle (Ozawa, et al., 2001). In all cases the property being maximised can be seen as the entropy production due to turbulent process in the system. Provocatively, Hisashi asked if MEP applies, do we really need to simulate the detailed structure of the general circulation of the atmosphere and ocean using general circulation models (GCMs)? Diplomatically, he answered both "no" if we are only interested in deducing temperatures and fluxes, and "yes" if we want to understand the mechanisms by which the system tends towards MEP and the range of possible states of the climate system. Some states may only be meta-stable, local maxima of entropy production, but the climate system may spend a significant amount of time in them. Hisashi's group have been exploring the multiple meta-stable states found in a 3-D ocean GCM and found that 'irreversible' transitions occur between them toward the most stable state, which corresponds to the maximum turbulent entropy production (Shimokawa and Ozawa, 2002). A cruder modelling approach gives the same MEP state but misses the other meta-stable solutions. The need for a certain amount of model complexity becomes clear when one considers that the ocean circulation has switched between different meta-stable states in the recent past, with far reaching consequences for the entire Earth climate system. In the long run, however, the system as a whole will evolve towards the MEP state within the limit of complexity accessible to it.

Toni Pujol (University of Girona, Spain) offered a synthesis for the two communities present at the meeting by applying the MEP principle to Daisyworld (Pujol, 2002). In the original Daisyworld, heat transport is parameterised by a parameter 'q' that represents the degree of insulation between surfaces with differing albedo. Toni freed up this constraint by allowing 'q' to assume a value that maximises entropy production due to heat transport. The result is that this gives a maximum (physically-allowable) range of survival of life on Daisyworld and the range of temperature regulation is extended. Toni has extended his model to one dimension with variation of solar forcing with latitude (in a simpler way than that presented by Ben Adams and Werner von Bloh in that only one type is allowed at each latitude). In this 1-D model the assumption that heat transport will adjust to maximise entropy production once again extends the range of regulation and survival of life. As is seen in other 1-D models, there is a high luminosity regime where life is restricted to 'hanging on' at the poles of the planet and has little effect on global temperature. One key assumption in Toni's work is that daisy area can vary to achieve a statistical steady state satisfying MEP. Thomas Toniazzo (Hadley Centre, Bracknell, UK) has considered the alternative approach of determining the heat transport from a MEP prescription at fixed daisy area. This gives a very different result: removing the regime of co-existence of the daisy types. Thomas went through a suite of possible approaches to applying MEP to Daisyworld, recovering Toni's result, but also suggesting that the relative timescales of different processes are crucial to determining the overall behaviour. He also noted that as MEP is being thought of as a statistical thing one really needs fluctuations in the model, and Daisyworld is too deterministic as it is. Existing cellular automata versions of Daisyworld offer a more stochastic framework for pursuing this research.

The final session of the day focussed on some of the different states of the Earth system over geologic timescales and the transitions between them. Ultimately, a principle such as maximising entropy production might help us understand how life has persisted for so long on Earth and why the Earth system spends more time in some states than others. An interesting example is the fact that the system has never got stuck in a fully-glaciated 'snowball Earth' state, despite the faint young Sun, and there having been at least two intervals (~2 Gyr ago and 0.8-0.6 Gyr ago) where low-latitude glaciations did occur. The conventional wisdom is that if the Earth freezes over completely there will still be a source of carbon dioxide from volcanoes, but no sink of carbon dioxide from weathering. Hence carbon dioxide will build up in the atmosphere until the greenhouse effect is sufficient to melt the equatorial ice and trigger deglaciation. Considering the entropy production of different states of the system offers a different perspective. The complete runaway glaciation of the Earth can only occur with a sufficiently vigorous latitudinal heat transport. Yet MEP suggests that as the Earth cools, heat transport will be suppressed, and this could prevent complete glaciation. In more general terms, a 'snowball' state is unlikely to represent a global maximum in entropy production due to heat transport. Thus it is not the most probable stationary state of the system, and one would expect a transition to a more probable, higher entropy production state without complete ice cover.

I reviewed the major transitions of the Earth system that have occurred since the origin of life. Perhaps the most important was the origin of oxygenic photosynthesis (>2.8 Gyr ago). This provided the biota with a sufficient supply of free energy to transform the composition of the atmosphere and the state of the surface of the Earth. Initially, a nearly oxygen-free, chemically reducing atmosphere rich in methane was created and maintained. Then, approximately 2.2 Gyr ago, a further major transition occurred to a chemically oxidising atmosphere with sufficient oxygen to form an ozone layer. The methane greenhouse effect would have all but disappeared and this could have contributed to the severe (low-latitude) glaciations of the planet that occurred about this time. Picking up on this topic, Werner von Bloh showed the results of a new model of the carbon cycle and global temperature. This predicts very high level of carbon dioxide in the atmosphere of the early Earth and hence high surface temperatures (because continents are assumed not to appear for the first 1.5 Gyr, hence with no weathering sink, carbon remains concentrated in the atmosphere). An interesting variant of the model considers two types of life: thermophiles that have high temperature preference and little effect on weathering, and eukaryotes that have a low temperature preference and a large effect on weathering (hence cooling the planet). The model system shows little co-existence of the two and a rapid transition from a thermophile biota to a eukaryote biota ~2 Gyr ago, with consequent sharp cooling of the planet.

Such a change and subsequent recovery of the system may provide an illustration of what Richard Betts (Hadley Centre, Bracknell, UK) and I have called 'sequential selection' for planetary self-regulation (Betts and Lenton, submitted). The idea is that should evolution produce a positive feedback at the planetary scale this can destroy itself, e.g. by pushing the system towards a snowball Earth, but it is unlikely to destroy all life, because there will be refugia in which other organisms survive. In contrast, should evolution produce a negative feedback at the planetary scale, this will (by definition) tend to persist. Thus over time, negative feedback stabilising a habitable state will tend to predominate. It was pointed out that this mechanism is similar to the 'ultrastability' introduced by W. Ross Ashby in his classic text 'Design for a Brain' (Ashby, 1952). Perhaps a mechanism such as sequential selection provides a means by which the Earth system can investigate the phase space of possible states, and ultimately 'find' the most probable macro-state that maximises entropy production.

The day concluded with a lively discussion of MEP and its potential applications. It was suggested that an attempt be made to extend Roderick Dewar's theory to cyclic steady states and time-dependent trajectories. It was noted that in chemical engineering the most efficient solution is sometimes a limit cycle, hinting that in some cases this could be the MEP state. There was much discussion of how MEP might apply to living organisms and ecosystems, many of the participants having been influenced by Schrödinger's classic book 'What is Life?' (Schrödinger, 1944). It was hypothesised that natural selection might enable living systems to be slowly refined towards an MEP state. At the ecosystem scale, Alfred Lotka had already noted in 1922 (Lotka, 1922) that living communities tend to increase their rate of free energy consumption (equivalent to entropy production) over what it would be without life and that this is a natural consequence of evolution - a more rapidly growing (more dissipative) species will displace a slower growing one. However, it was questioned whether ecological succession conforms to increasing entropy production, because succession is sometimes seen to end in a peat bog, which is not an obviously high entropy producing state. In contrast, the recycling of nutrients in an ecosystem, by boosting its capacity to take up free energy from sunlight, should tend to increase its entropy production. This might help to explain the remarkable degree of recycling of some elements in ecosystems, which is not trivial to explain from evolutionary theory. Moving to the planetary scale, discussion centred on the remarkable result that given sufficient degrees of freedom, an atmosphere-ocean-surface system should tend to adopt a state that maximises its entropy production. The Daisyworld studies hint that this has further important implications for life, because when MEP is assumed, the survival range of life (i.e. the lifespan of the biosphere) is maximised. It would be very interesting to know how general this result is, for example, is it contingent on life significantly affecting the heat transport (which it does on Daisyworld and to a lesser degree on the Earth)? If this result and the tendency toward MEP can be generalised it should make us more optimistic about finding life on potentially habitable extra-solar planets. Lively discussions continued as the participants tried to maximise their own free energy consumption and entropy production at the conference dinner.

In the final day of the workshop the focus shifted to considering new approaches to understanding and modelling feedback control systems. Peter Saunders (King's College London) began by showing how to build up an intuitive grasp of systems, using an example from human physiology. In collaboration with Johan Koeslag and Elmarie Terblanche (University of Stellenbosch, South Africa) Peter has put forward a new explanation of Type 2 Diabetes Mellitus. The body's ability to regulate blood sugar at precisely 5 mmol/l is remarkable, but can be explained by a mechanism of 'Integral Rein Control' (Saunders, et al., 1998) (analogous to the regulation in Daisyworld), in which this steady state is determined entirely by the competing effects of alpha and beta cells in the Pancreas. Alpha cells produce glucagon (which stimulates the release of glucose) whereas beta cells produce insulin (which stimulates cells to take up glucose). Regulation relies on there being effective communication between the two cell types. In type-1 diabetes the body can't produce insulin, this removes the fixed point of the regulator, and a new, higher set point emerges. In type-2 (traditionally "late onset") diabetes, the body produces insulin but not in the usual way (a series of pulses). The hypothesis is that the beta cells are not communicating efficiently and hence are not able to tell the alpha cells to switch off. Unfortunately, the alpha cells default position seems to be "on" (i.e. producing glucagon). In the early stages of the condition, the beta cells can cope and stop blood glucose rising too high (so-called "syndrome X") but with time the capacity to regulate deteriorates. Peter suggested some generalisations from this and his other studies of physiological regulation: that a totally connected system is not often stable, and that redundancy in systems control is good in the short term but may be bad in the long term.

The relevance of such lessons to ecosystems was taken up by Peter Henderson (Pisces Conservation, Lymington, UK) who discussed his observations of aquatic ecosystems in the flood plain of the Amazon basin and off Hinkley Point on the Bristol Channel. Pete opened by noting that field ecologists never believed Bob May's (1974) assertion that "in general, as a system becomes more complex it becomes less likely to be stable". His continuous data since 1980 from Hinkley Point show a dynamic food web subject to constant invasion by migrants that is both complex and surprisingly stable. Pete linked his observations of this real ecosystem to the predictions of the 'Damworld' model he produced with the late Bill Hamilton. 'Damworld' shows how a complex, stable ecosystem can be built up simply through random additions and subsequent selection. The model differs from conventional community ecology models principally through the inclusion of coupling between organisms and their environment - some organisms can build up (or destroy) a dam that creates a habitat for other organisms. Thus living organisms modify their environment and enrich the niche space for other organisms. The parameter space for stability may be small in the model but the selection process usually finds it. For the system to be stable it must also obey a few simple rules, for example, a jack of all trades cannot be a master of all trades. Co-operative pairs of species are particularly disruptive in the model, which may be because of the use of Lotka-Volterra population dynamics. Pete is currently working on including natural selection in the model, and it was suggested that a meta-population of Damworlds could be created amongst members of the research network.

Ian Marshall and Chris Roadknight (BT exact Technologies, Martlesham Heath, Ipswich) re-addressed the challenge of life detection, which led Jim Lovelock to the beginnings of the Gaia hypothesis in 1965. Rather than being charged with detecting life on other planets, Ian and Chris have a more localised remit. Hence they have been focussing on a statistical method of life detection based on pattern recognition (as opposed to Lovelock's planetary-scale thermodynamic approach to the problem). Their method uses a combination of pattern analysis (in the spatial domain) and wavelet analysis (in the frequency domain) of digital (jpeg encoded) images. An adaptive neural network learns to distinguish life-like from non-life-like patterns and processes. Thus far the method can distinguish wood from stone surface, but cannot yet identify once-living features in a rock (e.g. the stromatolites created by ancient microbial mat communities). There are many potential applications of the technology. One suggestion was to apply it to Peter Henderson's ecological time series data.

Marcel van Oijen (Centre for Ecology and Hydrology, Edinburgh) presented a cellular automata framework for modelling ecological feedback systems that he has been developing. Putting the original Daisyworld model in this spatial context generates some instability: with one daisy type on its own, oscillations can occur as the daisies spread to fill the space and take the temperature outside of the habitable range, then die back at the next time step (Lenton and van Oijen, 2002). It was noted that this is a consequence of treating the population dynamics discreetly with all the cells being synchronously updated. However, when mutation of albedo is introduced in the model the oscillations disappear. Marcel went on to discuss how the model is being developed and generalised to allow for factors such as spatial heterogeneity, a hierarchy of species, and the possibility of sexual reproduction.

In the final presentation of the workshop, Mark Staley discussed 'The market as a complex evolving system'. He noted that there is an interesting body of work on Darwinian selection amongst firms and individuals, with books such as "An evolutionary theory of economic change". Mark suggested such an analogy between economic and biological systems is reasonable and that therefore models from economics may be usefully adapted to address biological systems. This has already happened with great success in the field of Game Theory. As the economy contains something analogous to natural selection, the link to Daisyworld-type models is actually much clearer than for fluid flow. The challenge is to make a formal comparison of apparently different quantities, for example, what is the analogue to money in biology? It was suggested that it could be free energy. If so, does the economy do something analogous to maximising entropy production? Intriguingly, one study shows there is a 5/3 power law in both turbulence (which does tend to MEP) and in the stock market.

Existing models show that certain behaviours of complex systems can occur, but we are often more interested in how likely they are to occur. Hence the final scientific discussion focussed on how we can change our methodology to move from assessing what is possible to assessing what is probable behaviour of complex systems. Probable behaviour may be described by what has been called the (non-teleological) 'goal function' of a system. Goal functions usually take the form of an extremum principle and MEP is a good example. According to Roderick Dewar's theory, in non-equilibrium systems subject to local conservation of energy and mass and external controls, the most probable stationary state is the one maximising entropy production. However, we still need to clarify which systems of interest to us (e.g. organisms, ecosystems, Gaia) fall into this category. For those systems that do not, can the theory be extended to encapsulate them? And do they obey the same or different goal functions? Over 20 different goal functions have been suggested for ecosystems alone, usually with empirical support. Dimensional analysis may show that many of these goal functions are equivalent, but should we expect the number to ultimately be reduced to 1? In parallel with developing the fundamental theory, a top-down trial-and-error approach to deciding which goal function is appropriate for a given system could be more widely adopted. That is the method that was used in MEP studies of the climate system: assume an extremum principle and constrain a model system to obey it, then see if it predicts something reasonable in comparison to the real system. Alternatively, in a wider range of real and model systems, quantities such as entropy production could be diagnosed and the results examined to see if the system is tending towards an extreme state.

Our workshop showed that tangible connections between maximum entropy production, homeostasis, self-organised criticality and life-like dissipative structures are beginning to emerge from a range of studies of non-equilibrium systems. Thus the future for Gaia theory looks set to include a return to its thermodynamic roots.

Acknowledgements

I want to thank Peter Cox for first introducing me to Garth Paltridge's work on MEP, and all the workshop participants for their enthusiasm. The support of the EPSRC in funding the workshop and our Research Network in Systems Theory is gratefully acknowledged.

References

Adams, B. and Carr, J.: submitted, 'Spatial pattern formation in a model of vegetation-climate feedback', Nonlinearity.

Ashby, W.R.: 1952, Design for a Brain, Chapman & Hall, London.

Betts, R.A. and Lenton, T.M.: submitted, 'The origin of planetary self-regulation by means of sequential selection', Clim. Change.

Campbell, P.: 2002, 'Pursuing arrogant simplicities'. Nature. 416: 247.

Dewar, R.C.: submitted, 'Information theoretic explanation of maximum entropy production, the fluctuation theorem and self-organized criticality in non-equilibrium stationary states', Journal of Physics A.

Lenton, T.M. and van Oijen, M.: 2002, 'Gaia as a complex adaptive system', Phil. Trans. R. Soc. B 357, 683-695.

Lenton, T.M. and Wilkinson, D.M.: submitted, 'Developing the Gaia theory', Clim. Change.

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Lorenz, R.D., Lunine, J.I., Withers, P.G. and McKay, C.P.: 2001, 'Titan, Mars and Earth: Entropy Production by Latitudinal Heat Transport', Geophys. Res. Lett. 28, 415-418.

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Pujol, T.: 2002, 'The consequence of maximum thermodynamic efficiency in Daisyworld', J. theor. Biol. 217, 53-60.

Saunders, P.T., Koeslag, J.H. and Wessels, J.A.: 1998, 'Integral Rein Control in Physiology', J. theor. Biol. 194, 163-173.

Schrödinger, E.: 1944, What is Life?, Cambridge University Press, Cambridge.

Shimokawa, S. and Ozawa, H.: 2002, 'On the thermodynamics of the oceanic general circulation: Irreversible transition to a state with higher rate of entropy production', Q. J. R. met. Soc. 128, 2115-2128.

Staley, M.: 2002, 'Darwinian Selection Leads to Gaia', J. theor. Biol. 218, 35-46.

   
   
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