Nature's Holism (condensed - 10)
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Chaos, Holism, Nature


Complexity as a science has to do with structure and order that are to be found between the condition of total randomness (chaos) and total order. Complex systems are organised (Lewin, 1993). Our planet is a complex system. The significance of chaos and complexity is that this separate and independent route arrives at similar conclusions concerning ecosystems and compatibility. Where experimental confirmation of a theory is not possible, perceptions and principles arrived at via independent routes serves as valuable support for the validity of the idea. The terminology may be different, but the conclusions are the same.

Ecologists recognise that "Ecology is not a science with a simple linear structure: everything affects everything else" (Begon, 1986). Ecosystems for an ecologist and evolutionary biologist are dynamic processes rather than a state. Evolutionary biologists are interested in why the mechanisms are as they are to be found (Mc Farland, 1993). Why for example, does interdependence evolve within an ecosystem? The study of chaos and of holism requires turning away from reductionism, the analysis of systems in terms of their constituent parts (chromosomes, genes, DNA) and investigating the characteristics and behaviour of the WHOLE SYSTEM - the holistic approach. A living system has constraints or boundary conditions and it evolves. Life with evolution, as characteristics of the complex systems that we call ecosystems, provides the necessary conditions for the important mechanism of feedback. This feedback between associated organisms and their physical environment, results in the formation of systems and therefore wholes with interdependent parts. An ecologist has this approach. In the early stages of this theoretical development, it is necessary that the language be simple and not too specialised and that it remains so if possible. To study wholes, we first have to identify them. An organism is a whole - remove the heart and it dies.

Galileo, the father of modern science employed the empirical approach, combined with mathematics, in scientific experimentation. As some of the possible behaviours are neglected, such mathematical descriptions become approximations. Much of what we perceive is a model of the real. As Alan Watts noted in his book on Taoism, "Although thought is in nature, we must not confuse the game-rules of thought with the patterns of nature."

Galileo had to disregard friction and air resistance as "non-linearities" when he dropped a feather and ball off the tower of Pisa, yet in the real world they are very significant: the feather must float slowly to the ground and even be blown away by the wind. Einstein eliminated the effect of air resistance by comparing objects in a free falling elevator where all bodies would experience weightlessness and "viewed from the reference frame of an observer standing on the ground, these bodies (feather or iron ball) all fall with equal acceleration" (Davies & Brown, 1988). Yet without air resistance to consider, physicists still find two other factors to consider within the lift! Two falling bodies would exert a minute force of gravitational attraction between each other, but this would be arbitrarily small if the bodies were light. As falling bodies are drawn toward the CENTRE of a spherical earth, the two bodies fall on very gradually converging trajectories. Furthermore an object closer to the centre of the earth experiences a larger gravitational force than an object slightly further away (gravity diminishes with distance according to the inverse square law), so in free fall two such objects will tend to get further apart as the bottom object falls more rapidly.


The behaviour of apparently simple physical and non living systems becomes very complicated using the reductionist approach. Analysis of a dripping faucet using sets of coupled non-linear partial differential equations with suitable constraints or BOUNDARY CONDITIONS soon leads to the physicist finding himself "lost in a deep, deep thicket" (Gleick, 1988). A solution to this problem for the Chaos theorist is to ignore the physics of the system - the viscosity of the liquid and such like - and analyse the data of the drip. He analyses features of the whole system. Any data that he studies come from the whole system. This field of study, called chaotic dynamics, provides information on the behaviour of the system. Interactions within ecosystems are similarly highly complex due to the multiple species and constraints involved. Different kinds of organisms do not distribute themselves randomly amongst different types of environment (Begon et al, 1986). A study of nature is approachable from many levels, but the observations are approximations of the real system. Widgets

Chaos emerged as a science in the 1970's. Ecologists employ mathematical tools to describe the dynamic processes found in nature. Differential equations of mathematics describe the way systems change smoothly and continuously over time and so we may use them to study DYNAMIC SYSTEMS. A model is built as "a mathematical construct which, with the addition of certain verbal interpretations, describes the observed phenomena" (Neuman, 1988, in Gleick). How we perceive something is largely dictated by the interpretation. By setting basic parameters of an ecosystem model at the start of a time sequence we can study the consequence of variations. It is important to be aware of inherent limitations to such systems. Animals have an inherent maximum reproductive rate, providing a physical constraint to a specific dynamic component. A human mother cannot have twenty offspring in one year. Consequently, population growth rates are a function of existing population numbers. This year's population determines the limits to growth of the following year's population. Biological and physical processes are bound by real and natural constraints on possible behaviour. If the system under study is in some way a whole or a closed system, these constraints will limit the dynamic processes that are possible.

We can develop simple models to follow a hypothetical population through time. Each year's population depends on the previous year's so that the whole history of a population becomes available through this process of functional iteration. In a feedback loop, each year's output serves as the next year's input. In non-linear systems there is disorder and erratic changes, which reduce the possibility of deterministic analysis as time proceeds, yet the study of chaos led to the discovery and establishment of what is called, DETERMINISTIC CHAOS. "Chaos brought an astonishing message: simple deterministic models could produce what looked like random behaviour. The behaviour actually had an exquisite fine structure, yet any piece of it seemed indistinguishable from noise." Similarly, ecosystems are systems in motion, or dynamic systems in which specific species may appear to be engaged in a brutal struggle for survival of the fittest, but as a complex non-linear system, the ecosystem has a deterministic outcome (It is not however "self-determined"). Creatures evolve within the interacting constraints of perpetuity (population numbers and growth rate) and compatibility (ecosystem stability) and the outcome is a compromise between the two. Functioning within the whole, perpetuating through all time in association with the whole system, including other organisms, the constraints that an animal encounters have to be defined in relation to part of the whole with which it interacts. The animal that destroys its particular niche will have exceeded the constraints defining the association and become extinct. These constraints bind an animal's behaviour and physical form.

Models are always mere approximations of the real world. They have severe limitations. Regular and simple equations can produce irregular behaviour, pointing to both the limitations in such (mathematical) models and the potential for very complex behaviour in nature's intricate web of interactions. Population growth has limits that are inherent within the animal. Over time feedback occurs. This year's population will have a bearing on the size of the following year's population. Mathematics is used to explore the effects of natural processes in an attempt to make predictions such as the dynamics of fish populations for commercial fisheries, the expected course of disease epidemics passing through a population and so forth.

The formula: x(next) =  rx(1-x)

represents a mathematical formula supposed to model nature. r represents the population growth rate.

graf1b.jpg x represents population numbers as a number less than one, where one is the factor representing the maximum population numbers. Population number is expressed in this formula as a ratio between zero and one, where one is the maximum attainable number. The output from the formula becomes the input for the next cycle, so mirroring how a reproducing population behaves. Although the reproductive rate may be constant, the population numbers affect how many young enter the next generation.

I illustrate the behaviour of a population according to the formula in Fig. 1. (x starting at 0.03 and r = 2.9). A population is supposed to rise rapidly overshoot its environmental limits, fall back below its carrying capacity and oscillate over this limit until it reaches equilibrium.

graf2b.jpg When r is increased to three, (Fig. 2) the oscillations do not converge on a stable equilibrium, even after 100 cycles (iterations, generations).

graf3b.jpg If r is increased to 3.1 (Fig. 3) there is a definite separation of the oscillations into two "periods". This is precisely the same formula, with only the reproductive rate (r) increased!

 If r = 3.44 we see new behaviour emerging - the two periods begin to split (Fig. 4). graf5b.jpg

By the time r = 3.5 there are clearly 4 periods (fig. 5). Again the only variable in the formula to change is the growth rate (r) of the population. The population is oscillating between four levels in a regular, stable pattern.

To smooth out the data we now start with x = 0.5 and put r = 3.56 (Fig. 6). Again we see an increase in the number of periods. What is happening? We can interpret this by saying that it is possible that where resources are limited, this acts as a feedback limiting further population growth, and the population may be thrown into some very regular cycles or oscillations that vary with its reproductive rate and the carrying capacity of the environment!
graf7b.jpg Let's explore this further. By r = 3.6, order is starting to collapse (Fig. 7). (In nature changes are random and prevent such regular patterns. In statistics (e.g. regression analysis) they lump together these variables as  "stochastic  disturbance " or "stochastic error term" (Gujarati, 1988). This disturbance term makes allowance for factors that affect the system, but are not included in the model. Mathematical models are useful in showing the underlying patterns.
graf8b.jpg By r = 3.7 order is largely lost (Fig. 8).

The system is displaying chaotic behaviour because of the interaction of the growth rate (r) and the carrying capacity (limits) (K = 1) of the model! The specific resources of the niche determine the carrying capacity, so one can say that severe resource limitations cause the fluctuations.

By r = 3.99 (Fig. 9) the system governed by our simple formula has become chaotic - periodicity has given way to chaos, fluctuations that never settle down (Gleick, 1987). A biologist gathering such data from nature would never guess that there is a pattern to this erratic behaviour. How could he perceive that the same system, with only slight changes in some parameter, could display such a variety of patterns? Further, stochastic variations from other influences are not even considered here!

graf10b.jpg    At r = 4 the mathematical formula fails and the population collapses to zero. The mathematical "environment" could not cope with such a high growth rate, limits (k<1) were greatly exceeded and "extinction" occurred (Fig. 10)!


  Let us try to bring the model's expression closer to the reality of what occurs in nature and see how  chaos can occur. In fig 11 there is a slow incremental increase in the growth rate (r) of the population over time to illustrate all the above behaviour (Figs. 1 to 10) in one picture. From Fig. 11 we can see that there is an initial, rapid increase in population numbers. In this region, resources or the carrying capacity of the habitat does not limit the growth rate, but only the intrinsic rate of increase of the population. The organism has biological limits to the number of offspring that it can produce. Then the curve flattens as the effect of environmental limits comes into effect. Feedback from the limits that define the carrying capacity of the environment cause this. As the growth rate does not respond sufficiently to the responses from the environment, the animal's numbers explode into chaos near the carrying capacity of the environment. At one point extinction occurs. When the growth rate is very high, this happens very quickly. Here the carrying capacity is a value of one. Growth is smooth until they approach this value and then with sustained growth, suddenly the smooth curve bursts into chaos. This is the region that chaos theorists call the edge of chaos. The formula for this curve is:

x(next) = r(inc)x(1-x)

where r(inc) = previous r value + 0.005. With each iteration of the formula the rate of increase (r) increases by 0.005.

graf12b.jpg Near the carrying capacity, the growth rate (r) decreases. Clearly, the reproductive rate is subject to environmental constraints. Plant grazers, for example, would find less food available as their population numbers in an area increased over time. Thus as population numbers (x) approaches one, the reproductive rate (r) must decline. The animal causes changes to its environment and then responds to its environment (feedback). This environmental change takes on many forms, from available nesting sites to sufficient food for the population, to environmental pollution as we humans are discovering. By inserting this condition into our formula, we find that the population growth never explodes into chaos (Fig. 12). Here there is a rapid feedback from environmental conditions affecting population growth rates. This is more likely the common case in nature.

A situation can occur in nature where some resource suddenly disappears and a population's growth rate (r) or population density is suddenly too high. This can happen during periods of drought. It can occur in the oceans to important fisheries when temperature changes (e.g. El Nino) alter the availability of food resources or when commercial trawling reduces the food available to wild animals. Sometimes, birds, seals or other predators of fish stocks starve to death in large numbers. This happened to the seal population off the west coast of Southern Africa in 1994. In effect, a resource limit (environment), throws the population into chaos.
In 2001, Enron Corporation went bankrupt after a rapid sale of shares caused a rapid drop in share price. This is equivalent to the sudden decline of a food resource in nature, but in this case, the company became "extinct", as it was a single entity ("population"). What emerged were many smaller viable companies (or units) bought by other companies. A resource decline, be it food, or investor funding,  immediately influences the original growth rate, decreasing it. We see the impact of the environment upon the reproductive output (r) in Fig. 13. Chaos smoothes out to a new stable situation, the epitome of  holism. Note that there is no self regulation here, only the dynamics of living, responsive and dynamic systems! Here the initial growth rate was four, placing the population at the edge or in total chaos. I calculated this rate as:

r(next) = r + 0.005 - x/100. As the population numbers (x) approaches one, there is then a negative impact on the growth rate (r). This in effect mirrors the impact of the environment upon a population displaying chaotic changes due to a high reproductive rate.


Data from nature or a business that appears chaotic as in the beginning of Fig. 13 may be reflecting stress due to a growth rate that is too high in relation to the constraints (resources) of the environment. Nature quickly reduces the growth rate through a direct effect upon the reproductive output of each individual. If businesses do not respond to financial limits, through constraining their growth rate, they may go bankrupt (see Enron example )! In this graph, each point represents a generation, so the impact is slow, but very effective.

Chaos in  nature:

In this graph, each point represents a generation, so the impact is slow, but very effective. Figure 12 is quite an important graph. Initial conditions determine the outcome. In nature the response is often much more rapid. In times of environmental stability many variations of a specific trait may exist. Genetically, such a population is polymorphic (diverse) at various gene loci on the chromosomes in the nucleus. (A gene is a segment of a DNA molecule in the nucleus. A structural gene that codes for a polypeptide (protein) chain needs to be about 1000 base pairs long (see section on genetics) (Beck et al, 1991).) Under stable conditions, variations, or alleles as they call them when one is talking about the genes, which are not immediately detrimental, accumulate in the gene pool. An example is to be found in the buffalo of the Kruger National Park in South Africa. In the early 1990's a prolonged drought spanning almost three years caused the death of 14000 buffalo, leaving only 16000 animals alive! There is a high probability that the survivors were genetically predisposed to handle the drought while most of those that died possessed the wrong complex of genes to cope with this natural constraint. A three-year period is not enough time for a buffalo to attain maturity, so here is an example of extreme, but natural selective pressure that acted upon a wild population.


In aquaculture systems the carrying capacity is an important idea. This table lists some factors that might be critical in affecting how many animals a seawater pond can hold and therefore, its carrying capacity. These factors cannot be considered independently, but we must evaluate them compared with the biology of the organism being cultured. An organism's biology reflects its niche in nature, so what applies to aquaculture also occurs in nature.

Physical Parameters:
* Temperature range (daily and seasonal variability)
* Salinity range (tidal and seasonal variability)
* Particulates (solids, silt, organic matter)
* Light (intensity, day length, seasonal changes)

Chemical Parameters:
* Water quality 
- pH, alkalinity,
- Gases (oxygen, nitrogen, CO2, H2S, gas pressure)
- Nutrients (nitrogen, phosphorus, trace metals)
- Organic compounds
- Toxic compounds (heavy metals, biocides)

Biological Parameters:
* Disease causing organisms (bacteria, fungi, viruses, others)
* Food resources
* Predators
* Parasites

Habitat Requirements:
* Space (territorial, shoaling, benthic, sessile etc.)
* Life cycle (nesting sites, dispersal mechanisms etc.)

The above graphs condense many ecological principles. When considering the carrying capacities of a habitat, we are usually talking about the resources needed by the animal concerned and the constraints imposed by the specific environment (Table x). Ecologists tend to measure those parameters that they can most easily relate to the carrying capacity, such as available food or a change in the rate of a population's increase.

The carrying capacity (K) represents the population size that the resources of the environment can sustain. All things consumed by an organism are resources for it (Begon et al, 1986). For a plant, even radiant energy is a resource upon which it depends for energy (Begon, et al, 1986). Traditional ecological theory says that organisms compete for these resources. Plants in a forest compete for light from the sun. Competition is an interaction between individuals, caused by a shared requirement for a limited resource, reducing the survivorship, growth and reproduction of the interacting individuals. An animal occupies and is adapted to a niche upon which it depends for its various resources. From the perspective of population dynamics, the changes in reproductive output are a response to resource availability. Births (fecundity) and mortalities affect reproductive output. Competition is not a factor that has to be considered here. The term "interaction" is more objective.

A response caused by resource availability influences the individual organism. This is the approach or interpretation that I have applied to the above graphs. Their interpretation and understanding do not require "competition". The interaction is an organism-resource or organism-environment interaction. A plant seed that falls and germinates under a cliff will be light limited due to the shading effect of the cliff. We do not say that the plant is competing with the cliff. How the resources become limited does not affect the interpretation of the dynamics of the interaction. Traditional ecologists say that the "ultimate effect of competition is a decrease in reproductive output" (Begon, et al, 1988). The "resource limit interpretation" I use says that the ultimate effect of environmental constraints, through any process, is a decrease in reproductive output. Where scientists interpret density-dependent fecundity or mortality as "competition", environmental constraints are the actual factors causing these changes. Even if an animal is chased away from its food resource, the impact upon its reproductive output comes directly from the resource limitation and indirectly from an interaction between living creatures.

Chaotic systems are damped (yin) and driven (yang). The weather for example is damped by the friction of moving air and water and the dissipation of heat to outer space and driven by energy from the sun. A continual production of offspring drives ecosystems, while constraints conferring habitat stability, including mortality, damp ecosystems. Several processes acting as feedback to the impact of offspring entering the system cause this damping. A predator responds to excess prey by higher reproductive success and a greater predation rate. Such systems are dynamic, never in total equilibrium and are therefore "creative" processes. However to avoid any teleological implication, it is better to merely describe these as DYNAMIC PROCESSES subject to physical constraints. While the weather is as a PHYSICAL process, the chaotic dynamics of living systems are "evolutionary". Physical systems do not evolve, but in biological systems, long-associated species adapt to each other through evolution. Fundamental mechanisms of such systems may be understandable but the long term behaviour of such systems, being non-linear, cannot be easily predicted. The traditional method of looking at systems locally, then isolating the mechanisms and finally adding them, fails to predict the system's behaviour.

Chaos in  business:

"Asian Flu":

The material to explain chaos in business is evident in the 10 years of financial markets up to 1998. Many countries had seen good growth until then. Suddenly limits to the rate of growth caused a collapse in many countries. High flyers such as George Soros lost  US$ 2 billion in Russia! Time (Vol.153, NO6., 1999) reported that 40% of the world's economies changed from "robust growth onto recession or depression". All the symptoms of chaos are evident in this process of change. Market fluctuations became very evident, as in the above graphs. Ramo, of Time, observed that "the tiniest perturbation could send the whole (US) economy tumbling, and there were perturbations all over the place." In Asia this crash was particularly severe. In terms ofchaos theory, the rate of investment did not slow, and so the rate of growth did not slow, as the region's carrying capacity was attained. 

Now, what is the carrying capacity of an economy? Over 10 years until 1987, half a trillion dollars were invested into the Asian economies. As better investments became saturated, their share price rose and money was poured into "useless real estate and industrial development". With countries like Thailand growing at up to 13% per annum and Malaysia at up to 9.5%, investment by stock brokers looked like sound strategies.  The peak was reached in 1998 and then the system was on the edge of chaos and finally went chaotic. The effect was rapid. Growth slowed, investors withdrew their money and the crash followed. 

As is the case with all complex and potentially chaotic systems, Rubin of the US Treasury, noted that "Everything is probabilistic". World markets are described as volatile. With a knowledge of chaos, how could economists have averted this collapse? The solution would have been to evaluate a sustainable rate of growth each year and then to regulate the rate of growth to remain at the edge of chaos. Instead of the rate of growth being driven by investment, investment should be regulated in response to real economic constraints within each country. As there was not internal response to the saturation of the targets of investment, the system went chaotic. This idea is different to the conclusion of Greenspan and Ruben that "trying to defy global market forces is in the end futile". With a recognition of complex systems comes the principle that the rate of growth should be monitored and regulated to prevent a major correction.
At the height of the crisis in Asia, the IMF promoted increased interest rates so as to slow the movement of money and investment, but this strategy added fuel to the fire, as the system was then already chaotic. Their remedy was simply another factor causing chaos! The aim of the IMF is now to "reduce volatility". In terms of chaos theory, they need to keep the rate of growth under control and sustainable, so that growth is on the stable side of the edge of chaos. A form of control was promoted by economic thinkers such as Soros and Makathir of Malaysia. Their solution is to lock in capital investment, which is different to controlling the rate of growth. Market forces can still operate freely where the rate of growth is controlled. Locking in capital may however prevent natural corrections. 

An ominous conclusion to all this is that currently (1999), the US economy is "more dependent than ever on the stock market". (This prediction was realised in 2001 with the collapse of Enron Corporation.) Another is that the human population has increased from 1 billion of the 6 billion in the last 12 years. It took 200,000 years to reach the the first 5 billion! This is where chaos reigns! By the year 2000, Greenspan had changed the colour of his spots! In accord with the above chaos principle, he recognised that stock price rises should be in some way curtailed, suggesting that they should not rise more than the growth of the overall U.S. economy (less than 10%). This is heresy in an economy where investors often expect double digit annual returns on their investment. A strategy to achieve this slowdown of the American economy was to raise interest rates so as to bring the growth rate down from 6% to about 3%. Other Wall Street economists were "baffled by Greenspan's thinking" (Time March 20, 2000).

Enron, a classic example of chaos:

The rise and fall of Enron Corporation, from 1985 to 2001( see share price graph ), is a classic example of the forces of chaos impacting upon business. Terms used to describe this collapse include "total system failure", "implosion", "another bubble", "bubble exploded like a grenade", "a smoking ruin". By the time regulators and financial markets "finally reined Enron in", it was too late. In terms of chaos theory the exponential increase in growth rate resulted in growth exceeding available resources, entry into chaos and rapid extinction. Unlike the internet start-ups that had rapidly collapsed a few months earlier, also in accord with forces of chaos, investment in Enron was believed sound, as it had real businesses, assets and revenues (natural-gas pipelines, electricity generating plants and water companies). Capitalistic policy believed that the "efficient hand" of market forces would provide a natural self-regulation, yet, like the Asian market collapse and failure of the Internet dot-coms, these principles failed to achieve the desired natural regulation. These entities follow the same rules as nature, but following a policy belief that an unlimited rate of growth is possible, they all failed to see that constraints and limits in nature are also possible and necessary . The same principles apply to a predator and its prey. Predators limit their own numbers through territoriality. Territorial behaviour limits the rate of growth with increasing intensity as the predator population increases. 

At its peak, Enron was seen as an example of how to do business and one of America's most admired companies! With its fall, faults were revealed in many modern business principles or practices, such as "utilities deregulation", the measure of growth (viability) by a rising stock price, retirement investments (401(k)s) in stocks,  unlimited financializing without concern for earnings, "off-balance-sheet deals" and many more. Off-balance sheet deals allowed Enron to hide debt. Accounting manipulation allowed such deals to be left off the profit-and-loss statement of the company, so creating falsely high profits and not reflecting real debt. Both of these accounting tricks hid real, natural limits to growth from investors and so the company approached the "edge of chaos". Only some small stimulus was needed for the real limits to kick in, and the further the share price rose beyond the true, natural limit, the more vicious would be the natural response when it took place. In 1999, the company entered the chaotic realm, without recognising it (no chaos analysts were in their employ). This can be seen, clearly reflected in the share price fluctuation that occurred in late 1999, throughout 2000 and into 2001. Anybody with a knowledge of chaos theory would have seen the warning signs of an entity in the realm of chaos. A response in early 2000 when these fluctuations started, may have been soon enough to correct the situation and to unearth accounting irregularities. The very sharp peaks and rises are typical of  a graph reflecting chaos. From the perspective of the investor, there is little point in further investment, as the future of the stock is not only unpredictable, but unlikely to rise significantly in the long-term. Chaos is the realm of the gambler, not the investor.

In summary, all financial markets should be reviewed, with the examples of the Asian stock market crash, the dotcom failures and the Enron failure viewed through the lens of chaos theory, so that other so major collapses can be avoided. But, who is paying heed? Most stocks brokers still follow the litany uttered in a letter to Enron shareholders, "Enron is laser-focused on earnings per share, and we expect to continue strong earnings performance." (Newsweek, 21/1/2002, p28) In the end, revenues and profits proved unsustainable. (See the full analysis of Enron Corporation's 2001 fall through the lens of chaos theory).

Future trends:

Managers at banks, brokerage houses, and hedge funds have to remain bullish in the face of uncertain markets. To a large extent, their jobs depend on remaining positive and securing further investments. These leaders of the investment world need to return to basics. Wall Street needs to again use a company's "book-value" as an analytical tool . A book-value reflects some natural reality that can bring share prices back down to earth. Using Enron as an example, I will explain.  In the second-quarter of 2001, Enron's book value was $13 a share, but, excluding goodwill, it was only $9 a share. Goodwill is a balance sheet entry on the asset side of the ledger, accepted when the investors overpaid when buying out another company.  At one stage Enron' shares were ten times this value! When a company is purchased for more than its book value, the excess is placed on the asset side of the ledger as goodwill. Although goodwill cannot be sold, it does inflate the price of the acquirer's book value! As such most reflected book values are in themselves an inflated reflection. In Enron's case, the book value was overstated by 44%! In the below table are some other book values (as of 12 November 2001), (Puetz, 2001).

Stock Stock Price Book Value
Alcoa 34.99 13.24
Citigroup 48.30 10.64
Coca Cola 49.22 3.06
Walt Disney   18.95 2.59
Exxon   40.25 9.13
General Electric 40.41 5.04
IBM 114.08 10.65
Johnson & Johnson 59.56 3.11
Merck 64.61 2.43
Microsoft  65.21 5.37
MMM 111.54  14.12
Phillip Morris 46.83 -.69




Countrywide financial

For a quick view of the health of these companies, see the full analysis of Enron Corporation's 2001 fall through the lens of chaos theory.


For some, chaos forms a subset of a larger science, that of complexity. "You are dealing with non-linear dynamical systems. In one case you may have a few things interacting, producing tremendously divergent behaviour. That's what you'd call deterministic chaos. It looks random, but it's not, because it's the result of equations you can specify, often quite simple equations. In another case INTERACTIONS IN A DYNAMICAL SYSTEM GIVE YOU AN EMERGENT GLOBAL ORDER, WITH A WHOLE SET OF FASCINATING PROPERTIES" (Lewin, 1993). This sounds similar to Kantist teleology. In ecosystems, the individual organisms are components of the system that interact locally to varying degrees. Because of these interactions and the dynamic process of natural selection, a global property for the system emerges. The main property elaborated in this book is the emergence of compatibility. THIS PROPERTY OF COMPATIBILITY SYMBOLISES A FORM OF FEEDBACK that INFLUENCES THE BEHAVIOUR AND FORM OF THE INDIVIDUALS THAT ARE INTERACTING AND EVOLVING. Order arises out of a complex dynamical system due to the constraints and limits of the system.

The aggregate behaviour of individuals of the ecosystem results in the emergence of compatibility and the associated stability of the ecosystem. What is emerging is the result of constraints upon the living, evolving components of the system. Those animals that deviate from the dynamic constraints are rapidly eliminated through death and so fail to perpetuate. Predation rapidly removes a conspicuous bird fledgling, a spell of cold removes from the process of life those whose tolerance to cold is not good enough, the seed that fails to disperse to a suitable site or the young lizard which fails to find suitable food and the man on an island who destroys all his fuel resources, all die due to an inability to meet the constraints of the system. Much random chance (stochastic error ) is involved, but systems that evolve to eliminate chance events have a better chance of survival. As the system comprises living and nonliving components, natural selection leads to the survival of associated species as an inevitable consequence.

Within chaotic systems have been identified what are called STRANGE ATTRACTORS or simply, attractors. This idea of a strange attractor is defined in relation to another abstraction called PHASE SPACE. Theoreticians use these concepts as a tool to study the behaviour of whole complex systems. In a system of moving parts, say a mechanical or fluid system, we reduce the essential criteria to a picture, such as the x-axis and y-axis co-ordinates of Cartesian of graphs, based on numerical data. The model summarises the total state of such a dynamical system by a single point in this picture called phase space. This point will possess co-ordinates to plot it on the phase space. At each instant the point moves and the co-ordinates change, to represent the CHANGED CONDITIONS of the system as defined by the parameters of the phase space. Co-ordinates plotted on the phase space trace the behaviour of the system. A pendulum can display chaotic behaviour. Its attractor is a single point toward which the trajectory spirals inwards and comes to rest. It represents a steady state. "By definition ATTRACTORS HAVE THE IMPORTANT PROPERTY OF STABILITY. In a real system, where moving parts are subject to bumps and jiggles from real-world "noise" (stochastic variation), motion tends to return to the attractor. A bump may shove a trajectory away for a brief time, but the resulting transient motions die out" (Lewin, 1993). There are constraints upon the dynamic behaviour of the system. The pendulum, as a nonliving system can display quite deviant behaviour, but living systems quickly die if their behaviour deviates too far from the norm - as if the pendulum is tied to a fragile string that snaps if the behaviour is too erratic.

The analysis of data, which reflects the characteristics of a dynamic system, using the tool or notion of phase space reveals strange attractors for all kinds of systems. STRANGE ATTRACTORS SERVE TO DEFINE THE BOUNDARIES TO THE POSSIBLE BEHAVIOUR OF THE SYSTEM. The data of different systems have to be "embedded" within a phase space with enough dimensions. This method of chaos analysis reveals a deterministic and patterned structure upon the axes of the Cartesian graph representing phase space.

In the holistic ecosystem, interacting, interdependent organisms, depend upon a stable habitat for their perpetuity. For example, destruction of large tracts of rain forest leads to the extinction of many species.  Behavioural mechanisms leading to compatibility cause this stability in ecosystems. Living animals must contend with boundary conditions or constraints - if a resource such as an edible bird is over exploited it may be driven to extinction - to the detriment of the animal that depends upon the food resource. As an example of the practise of compatibility is a custom of a New Guinea tribesman who explained: "It is our custom that if a hunter one day kills a pigeon in one direction from the village, he waits a week before hunting for pigeons again, and then goes in the opposite direction" (Diamond, 1991).

Every component of the dynamic ecosystem that can change independently is another variable, or in statistical terms, a degree of freedom. The multiplicity of variables within this complex ecosystem creates a form of "noise" creating apparently random and transient trajectories or motions; yet two requirements guide the process that creates a biological form of the strange attractor: perpetuity and compatibility. Just as the chaotic pendulum swings over the strange attractor created by gravitational pull, so, the perpetuity-compatibility association constrains the natural interactions between organisms within ecosystems. A spring drives the pendulum and friction damps it. The continual input of energy from the spring keeps the pendulum in motion. Similarly, the production of offspring provides the input to the dynamics of the ecosystem - the continual productive "force" applied to the system. Gravity as a constraint, defines the attractor described in phase space, as a single orbit forming a closed loop in such driven systems. If the "initial conditions" which define the attractor are of low energy, the pendulum is stationary, forming a second attractor that is a fixed point. Ecosystems have to be driven to be alive, so this latter fixed state does not apply to ecosystems - the system cannot come to rest, for this would be death.

In the pendulum example, the history of the "system time"  within phase space is represented by the moving point that traces its orbit through phase space with the passage of time. Two variables represent this phase space (in this instance), [1] position and [2] velocity on a simple Cartesian graph with a horizontal and vertical axis. Such periodic motion as a driven pendulum would display a circular loop on the phase space "portrait". Complex systems such as ecosystems would have to be represented on a phase space of many, even infinite dimensions. This provides the problem of the phase space portrait having less clarity than the real system, so the ability to generalise and abstract is lost. By definition, if a multidimensional phase space was to represent an ecosystem, any point on that phase space could represent a possible behaviour of the ecosystem. However many representations (or states) of a variable would be transient: an extreme temperature experienced would return to the characteristic temperatures for the specific latitude of the ecosystem and not remain in the extreme range over the long term. In other words, inputs into the system do not change the attractor, but the behaviour around the attractor (Cohen & Stewart, 1994).

In the long term, the attractors constrain the possible behaviour of the ecosystem. Cohen and Stewart (1994) note that attractors are emergent phenomena in dynamical systems. The PROPERTY of stability contained in attractors determines that the behaviour of the ecosystem returns to the attractor! If the pendulum is disturbed by being bumped out of its normal trajectory its swings out erratically. Deviation from the normal pattern is constrained and in time returns to the behaviour of the normal pattern. Note this return to a normal pattern in no way attributes intention or teleology to the pendulum. It is a natural PHYSICAL PROCESS and the return to a stable state. Ecosystems behave in a similar way. The characteristics of the system, such as the length of the pendulum defines the limits to this behaviour. Creatures within an ecosystem will exhibit their own phase space defined by the constraints to which they are adapted (Cohen & Stewart, 1994).

We should define the continual and final state of an ecosystem by an attractor with a simple set of properties, stable, low dimensional and nonperiodic. This is the perpetuity-compatibility (pc) attractor. An evolutionary process forms it. Such an attractor is plotted in an x-dimensional phase space defined by the x-independent variables. Compatibility is similar to gravity in the pendulum example, providing constraints to the system and defining limits to the system's behaviour. Perpetuity is similar to the energy input from the spring, keeping the pendulum in motion and overcoming friction. This mortality rate would be similar to friction as it has a damping effect upon perpetuity.

Dependence and interdependence of organisms within an ecosystem results in real constraints: the number of worms in an apple is limited by the size of the apple; the number of bees in an area depends upon the number of flowers (ask any beekeeper); the productivity of many a flowering crop depends upon the number of bees (ask any fruit-farmer). Most constraints are negative feedback responses to the environment that lead to a decrease in the reproductive output (r) through resource limitation. Compatibility revolves around the energetic costs of interactions. If you ask the question, "What are the long term effects of natural selection upon the energetic costs of interactions," you will arrive at the formulation of compatibility. A study of the evolution of the robustness and engineering design of the limb bones of terrestrial animals will show a trend in increasing energetic efficiency requiring lighter limbs that function better.

In the pendulum example, velocity and position are used to plot the points in phase space. Position is arbitrary, but has a relation to the centre of gravity, the position of absolute damping without the pendulum being driven. In this analogy, absolute compatibility would infer no interaction and thus no "cost". The compatible variable in this analogy must vacillate around this point - the "position" compared with "absolute compatibility". Interactive effects between species can be neutral (absolute compatibility), negative or inhibitory, and positive or beneficial. Compatibility is the result of an interaction. Without an interaction there is no compatibility to be measured. One thus needs a way of measuring relative interactive effects. In some way, we require a measure of Compatibility. Perhaps there is more than one variable required here. The compatibility value is a relative value. Individuals of the same species need to interact to perpetuate the species, so we need to balance the interactive cost against the interactive benefit that results in survival. Such a value has to be a statistical average or a probability: the average value for the population. (Probability theory was introduced byLudwig Boltzman) . If we allocate such intraspecific (interactions between individuals of the same species) interactive costs a value of one, then we can rate interspecific (interactions between individuals of different species) interactive costs a value relative to this. I will deal with this further in the section where I develop the modified energetic Lotka-Volterra model (MELV model) to illustrate compatibility.

There is the problem of how to measure compatibility. Compatibility leads to ecosystem stability through coevolution and interdependence. As the pendulum behaviour is plotted in phase space, so the species' behaviour needs to be plotted in some phase space that will define an attractor. We have already established that an interactive measure, the degree of compatibility is similar to the position used as a measure for the pendulum. The measure of velocity, a reflection of the energy input from the spring, must in some way be related to a measure of what I have called perpetuity or the rate of change in population numbers of the species. An ideal situation would perhaps be maximal reproductive output and maximal compatibility. However, as reproductive output increases, compatibility generally decreases.

Lovelock, the originator of the Gaia theory, reacted violently to the idea of strange attractors in chaos. He calls it demonology, (preferring his Gaia Demon), a fashion, and a fascination with a mathematical pathology. As analogies, he says they are like black holes, like time bombs and that they parasitize the world. When investigating the reason for this, it seems that he misses their inherent character of stability, explanatory power and ability to define limits - the boundaries that he so likes. Chaos theory explains all his Gaia postulates in mathematical terms, with strange attractors as very powerful and impressive concepts. As in ecology where two species cannot occupy the same niche, so with ideas, it is either strange attractors of chaos or Lovelock's Gaia. Like a protective mother he had to lash out fiercely.

In studying complexity and its relation or relevance to ecosystems, the question of fitness arises. An image called fitness landscapes was developed to aid understanding. This idea was first introduced by the geneticist Sewell Wright (University of Chicago) in the 1930's. The idea is similar to geographic contour maps where a line joins areas of the same height. From a genetic point of view, different combinations of genes will have different degrees of fitness. These combinations are selected for through the process of natural selection, with unviable individuals dying out. The fitter individuals form the peaks (more surviving offspring) on the genetic fitness landscape. Relating this to the chaos theory above, the fitness peaks represent a combination of genes that relate to the characteristics of the organism suited to the constraints or boundary conditions of the specific habitat. Such a landscape would then represent fitness probabilities, with peaks of different heights separated by valleys. If fitness is measured numerically, the valleys would represent the few surviving individuals of the species with this genetic combination. Peaks represent the numeric superiority of the most successful genetic combinations. "The fittest of the packages has the highest peak" (Lewin, 1993). Animals live in a dynamic environment, subject to natural selection, so to cross from peak to peak is not possible as it would require a lowering of fitness (into the valleys) through genetic combinations that are being selected against through natural selection. Peaks represent surviving animals bound by constraints maintained by the process of natural selection. The peak is the interactive response to the dynamic system with which the animal is associated. Mutation or variation and natural selection maintain the adaptive peaks. One thing he doesn't note is that individuals from two separate peaks may interbreed, so mixing completely separate, good traits. These do not necessarily produce inferior offspring.

Fitness landscapes and the ideas of complexity are useful in that we reach similar conclusions to the perpetuity-compatibility concept of ecosystems. Stu Kauffman considered individual species, where he considered a fly and a frog, both with their own fitness landscapes and they INTERACT. He called these COUPLED LANDSCAPES, which change, "each responding to the other" as would be the more real situation in nature. He imagines adaptations of the fly to evade the frog and the frog to catch the fly - a form of coevolution. In his eyes, this is classic biological arms race where predator and prey constantly try to be one-step ahead of the other. This has been called the Red Queen effect (from Alice in Through the Looking Glass where they have to keep running to stay in the same place). "Species do not lead isolated lives but instead are linked inextricably with others. The evolutionary success of one species may therefore be as much a function of what other species do as what the species itself does. Some biologists go as far as to argue that the Red Queen is the driving force in evolutionary history, with environmental change playing only a minor part" (Lewin, 1993). WHAT IS IMPORTANT TO NOTE HOWEVER IS THAT AN ANIMAL INTERACTS WITH ITS ENVIRONMENT, BIOTIC AND ABIOTIC, AND THAT NATURAL SELECTION ACTS IN THIS PROCESS FORCING THE ANIMALS TO ADAPT TO THE INTERACTIVE REALM, BE THIS A CONSTANT OR DYNAMIC, BIOTIC OR ABIOTIC INFLUENCE. AS THE CONSTRAINTS OF THE DYNAMIC SYSTEM CHANGE, SO THE ASSOCIATED SPECIES HAS TO CHANGE. THE DARWINIAN PROCESS OF NATURAL SELECTION ACTING UPON NATURAL VARIATION WITHIN THE POPULATION PROVIDES THIS CHANGE. The survivors evolve within the context of the whole.

Stu Kauffman next considered a community of interacting species. "We tune the interactions - internal, between the genes in the species, and external, how one species impinges on another - we watch how the system works, how the average fitness changes with different combinations of interaction. . . . The system moves through activity states, maybe frozen, may be chaotic, but eventually it comes to rest, with fitness optimised, poised at the edge of chaos . . . . COLLECTIVE ADAPTATION TO SELFISH ENDS PRODUCES THE MAXIMUM AVERAGE FITNESS, EACH SPECIES IN THE CONTEXT OF OTHERS" . . . "the community settles to a position of maximum sustained fitness. Some of the species would be hovering in a kind of evolutionary equilibrium while others among them engage in Red Queen antics; but all are components of a system delicately poised" (Lewin, 1993).

Having come this far, the agents of complexity concede to unknowns: "We need a reason why biological systems become MORE COMPLEX through time. It must be very simple and it must be very deep" (Stu Kauffman in Lewin, 1993). "The science of complexity proclaims it (that there is something deep about the source of order in nature) to be true quite generally in the world. And yet no one can say exactly HOW the ORDER emerges, only that it seems to in your model systems. There's still a leap of faith, isn't there, that all this applies to the real world?" (Lewin, 1993). "Some systems tend toward order, not disorder, and that's one of the big discoveries of the science of complexity" (Stu Kauffman in Lewin, 1993). It is the modified energetic Lotka-Volterra model (MELV model) used in this book that provides the answer to the question of "how" which is posed and eliminates the need for a "leap of faith". The mechanism by which increasing complexity progresses and stability emerges is explained.

Finally, then, chaos theory enables a new interpretation for natural systems. This approach is not reductionist, but is a method of analysing the data of dynamic systems such as ecosystems to identify some type of deterministic outcome from the chaotic behaviour. From a deterministic model with strange attractors, arises a degree of predictability (Gleick, 1988). Chaos theory may serve as a useful tool in the study of holistic systems. This new approach to studying nature may reveal new explanations for the dynamics of natural ecosystems. Dozens of interacting and associated species fluctuating in an apparent random and chaotic manner may be understood in terms of strange attractors within the system.

Complexity introduces the idea of emergent properties (Lewin, 1993) characteristic of the larger system - properties of a holistic nature. Such systems display the natural emergence of self-organising dynamics . Kant also emphasised the self-organising character of living systems. However care has to be taken to correctly identify what are emergent properties. Such properties must be real, identifiable features, not open to interpretation and speculation. One part of the theory of complex systems is the chaos theory. We can describe a system such as nature, composed of non-linear complex systems, by a set of simple equations, the outcome of which diverges dramatically into chaos (Kauffman, 1993, in Lewin). The other part, which assumes the title of complexity, considers systems such as biological systems that do not diverge, but produce convergent flow and structure . Complexity is then the theory of complex adaptive systems (Kauffman, 1993, in Lewin).

Continue with the condensed version.

Non Biological Stability:
The model can also generate patterns not found in nature .

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by Laurence Evans 1998 - 2008


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