The success of physics in predicting natural phenomenon brought with it the rationale that nearly everything could be explained mathematically. Accordingly, mathematics has been utilized in nearly every field of science, both soft and hard. The field of ecology has adopted mathematics in a wide variety of forms. Since the mathematics of physics is based on the microscopic, on the conceptual and reduced, many of the applications in ecology have involved significant approximation. Unfortunately, the extent of approximation required by the complexity of the ecosystem is often prohibitive. Inspired by success of such fields as classical ‘macroscopic’ thermodynamics, have been several attempts to mathematically and physically generalize observed ecological phenomenon. The manifestation has been the extermal principle. To several ecologists, the goal has been to find holistic principles or macroscopic ecological ‘forces’, to which natural ecosystems self-organized.

Although these principles have not been grounded in microscopic theory, they are founded on years of observation and experience. They are a collection of intuitive insights by the experts in the field. In this sense, it is clear that they should not be used as algorithms or combinations. Rather, natural ecosystems can be evaluated through an aesthetics approach by their predispositions. The more successful ecosystem will likely have the greater potential for growth and, more importantly, development (survival implicit). Thereby, ecological extremal principles can more appropriately be considered positional parameters. These positional parameters measure ‘to what extent’ an ecosystem configuration may be predisposed to grow and develop.

The need for positional parameters in ecological modeling can be clarified through analysis of the predator-prey dynamics described by Lotka-Volterra type equations. Although the following non-dimensionalised model,

dz/dt=z[-1+s y/(1+y)]

is among the simplest in describing an ecosubsystem, incredibly complex behavior can result. The coefficient space can be partitioned through stability analysis.

There are essentially three regions in coefficient space:

Coefficient Space

The boundary between region one and two is the line g =a b (s — 1) while the boundary between two and three is g =a b (s — 1)/( s +1).

Region one represents the coefficient configurations which will lead, unavoidably, to erosion. Number two admits globally asymptotically stable solutions, while number three exhibits limit cycle solutions. If the model is configured to region one or two, the steady state solution can be deterministically calculated. However, if the model is organized to region three, there are no equations describing the exact solution. In other words, it must solve it numerically. The differential equations are repeatedly calculated, as difference equations, until the end goal is attained (e.g. the slate of the system one hour into the future). This is precisely the dilemma of the mathematical ecologist. An, set of equations detailed enough to adequately represent reality can not be solved analytically. Furthermore, many systems are so complex to defy numerical calculation. For instance, weather modelers can fully predict the state of the atmosphere at an airport one hour into the future. The only problem is that it takes two weeks to calculate!

Although it is beyond the scope of this paper, it can be argued that, given certain limitations (Heisenberg’s uncertainty, the finite speed of light, etc.), it is impossible to predict the indefinite future deterministically. Therefore, the future must be met as it comes, recursively.

This type of phenomenon can be found almost everywhere. Chess computer games offer a succinct example. The program evaluates the entire set of first moves searching for the satisfying choice. Then, it evaluates all the possible moves from all the previous possible moves. Then once again makes the satisfying choice for the first move. This choice which may very well be different from the original! If the program could evaluate moves up to checkmate, then the system could be considered determined. However, this is often not possible and the program is forced to choose material and positional parameters by which to navigate.

Similarly, the ecologist does not have the grand ‘equation of life’, and, assuming it did exist, could then not possibly conduct it’s simulation to indefinite ends. Therefore, material and positional parameters (e.g. indices, tendencies, hunches) must be the guides. There are two reasons why this recursive process is necessary in ecology. First, the flow of ecosystems is non-Markovian (i.e., present events are intimately coupled to those in the next moment in time). Therefore, and second, future periods of time in the model will produce state configurations which the modeler might not have expected. No matter what hunches or indices with which he was working, these new states might suggest a different predisposition than the initial states. Given this complexity, the recursive approach is unavoidable.

Laplace would not have liked this at all, for he believed that given the positions and velocities of all the particles in the universe, the ignite future could be calculated. The ecologist, along with the rest in the world, must be satisfied with the ‘blind’ progression through time. The poet C. Strom put it clearly, "you leap into the world of uncertainty with the belt of beauty around your waist."

The predator-prey equations described above permit a wide range of steady state solutions. Therefore, which should be chosen? This is a fundamentally important question and perhaps could best be answered with another question. Which are the wrong choices? The unfeasible realm in coefficient space is region number one. Regions two and three represent stationary state solutions. The ecologist’s first step should be to demarcate the unfeasible realm thus reducing the range in which the coefficients may exist.

Another way to look at it is in terms of von Neumann’s turnpike theory. The steady state solutions to region two and three are controlled by what mathematicians call ‘attractors’. In region number two, the attractor is stable in phase space and thus exhibits an globally asymptotic stable solution. The attractor in region three is unstable (node or spiral) amidst a confined a set such that periodic oscillation results. Independent of the initial conditions, the solutions win find there way, through unique trajectories, to this attractor. The attractor can be considered the ‘turnpike’. It is a semi-conditional state of the solution, one which can be thought of as the ideal, unhindered from the starting point (initial conditions) and the end goal (e.g. optimality conditions).

In this model, the feasible realm can be defined as that which has at least one (barring the knell solution) attractor or turnpike (more complex models with more intricate attractors, appear chaotic, or aperiodic). This is easy to accept, for without a turnpike there may be no semi-conditional valuing scheme. If this is the case, perhaps its not even a system? In the attractor-less region (number one) of the above predator-prey model, in fact there is erosion.

The next step in the aesthetics model must be to address the range of solutions in the feasible realm. If for instance, there was some condition which always had to be met, e.g. E=mc², then most likely one solution would rise above the rest (actually, all but one would fall). However, as discussed earlier, this is not the case and alternatively, positional parameters can be used as guides. An important trait of this type of modeling is that the coefficients are not fixed through time. Such algorithms have been referred to as goal functions.

Positional parameters mimic the intuitive process of an expert in the field. Therefore, it would be proper to begin with an expert’s observations. E.P. Odum (1969) published 24 traits describing ecosystems. The overriding quality of these traits are that they all focus on the developmental direction of ecosystems. These can be used to characterize an ecosystem as either young or mature.

Positional parameters can be split into three categories: attributes, structure parameters, and characteristic parameters. Space and time could be defined as attributes. The structural parameters could be indices like nutrient cycling rates (#17), ratios of unit biomass supported to Input energy(#3) or standing crop biomass to gross production(#2). Finally, characteristic parameters could be such traits as specialization(#l2), equitability(#9) or feedback control(#l8).

The idea behind Odum’s indicators was that the development of an ecosystem would be most visible in its early to mature stages. The ecosystem doesn’t stop developing but it is often more noticeable in early stages. An ecologist attempting to predict the changes, or direction of a natural ecosystem may favor those which tend to increase, for example, specialization or cycling of nutrients. Any changes to the system (e.g. in growth rates) which would increase specialization or the cycling of nutrients may be more likely than those which would, for example, decrease specialization or the cycling of nutrients. Therefore, once the feasible realm has been ideated, the coefficients might be adjusted so as to bolster a predisposition indicative of developing systems. For example, increasing the uptake rate of the heterotroph in the model might increase the cycling of nutrients. If this is the case, then perhaps the coefficient change should be made, all else being equal.

Maximum Power Principle:

H.T. Odum eventually unified several of E.P Odum’s observations (or in aesthetics modeling, positional parameters), to a single extremal principle. This was the Maximum Power Principle. Coma actually postulated it earlier, in 1922, with the hypothesis of the maximum flux of energy, in attempt to describe the underlying rails of evolution. The Maximum Power Principle has come to include the maximization the throughput of emergy. Emergy is the embodied energy of a type of energy. For example, if 100 units of solar energy is utilized in the growth of plant with a 10% absorption rate, 10 units of energy would be tied up in the resultant plant biomass. However, the energy is more complex than the solar energy and the weighted value is the emergy. Therefore, the plant growth contains 100 units of embodied solar energy. Suppose that a cow grazed the 10 energy units of plant at 10% absorption. Then 1 energy unit would be associated with the growth of cow biomass. However, the emergy of this biomass would be 100 units of solar energy. Odum hypothesizes that ecosystems self-organize to maximize the throughput of emergy.

The ecological modeler using the Maximum Power Principle alters one or more of the coefficients to maximize the flow. For example, in the predator-prey described earlier, it can be shown that altering the maximum honest rate of the predator (hidden in sigma), leads to maximum flow of emergy for the configuration on the border of region two and three in coefficient space.

Maximum Exergy Principle:

Another extremal principle is the Maximum Exergy Principle, developed by S.E. Jorgensen (1982). The Principle states that the energy of an ecosystem is defined as the thermodynamical information (or negentropy) I multiplied by the absolute temperature T:

Ex = I * T

The thermodynamic information of a compartment in an ecosystem can be expressed as:

where Seq is the entropy of the considered system under thermodynamic equilibrium and S is the entropy of the system under the prevailing conditions. The thermodynamic information is a measure of the molecular order of the system. The information is indicative of the extent of organization. Although Jorgensen recently published a method of calculating the exergy based on the gene structure of the organism, h can, in some practical cases, be considered the concentration of biomass. The principle says that the system will select, from the available gene pool, those genes which contribute most to the exergy of the total system.

Environs:

B. Patten developed this principle through inspiration by von Uexkull’s function circles theory. Patten thought to focus on the relationships of the organism to the environment.

In this way, the entire universe of the organism is defined through the closure of the output environment to the input of the environment. There is an infinite connection between the organism and the organism-environment resulting in a synergistic relationship.

Patten based the quantification of ‘direct’ and ‘indirect’ effects within an ecosystem on what he named an environ. For example, consider the four compartment model in Figure 2.

The quantification of environs requires the counting of the number of interrelations between the compartments. Between the microbes and invertebrate #2, there is one direct effect (length one path). However, there are three length two indirect effect paths from the microbe to invertebrate #2: through invertebrate #1, from microbe to microbe to invertebrate #2 and from microbe to invertebrate #2 to invertebrate #2. Each of these connections is defined as an environ. They can be simply counted using matrix algebra. Due to the loops and cycles the number of paths diverges with greater path length, however, the influence converges. The influence is calculated by replacing the direct path lengths with the fraction of inter-compartimental carbon flow to compartment carbon stock. Patten hypothesizes that ecosystems will evolve so as to increase the number of environs. Therefore, the related edemas principle is the maximization of indirect effects to directs effects.

Ascendancy:

Thermodynamic systems are defined by forces and fluxes. R. Ulanowicz has concentrated on the flow of biomass through the system. Classical thermodynamicists (e.g. Carnot, Clausius, Kelvin) concentrated on the flux of the chemical species rather than the force, or chemical potential. In equilibrium, the chemical potentials equalize and the flux is trivially zero. However, holding the system away from equilibrium(maintaining a gradient of chemical potential) will result in macroscopic flux. Although no ecological biomass ‘force’ has been found, the associated systems are far from chemical potential equilibrium and therefore admit macroscopic flux. Ulanowicz has accordingly attempted to understand ecosystems, inferring the effect of the forces, through the biomass flows.

Growth in this case is defined as an increase in throughput, T. of biomass in the ecosystem. This is closely related to Odum’s Magnum Power Principle, however, there is no valuing of the varying complexities of energy. Given two compartments, j and k, there is communication through the means of the flow. Thus, for compartment j to communicate with k means that it must contribute to the size of k. Therefore, there is a mutual sustenance maintained between the compartments, including such concepts as positive feedback and autocatalysis. Throughput of the system takes the forth of individual flows, Tj. The fraction of flow, Qj through any compartment j is such that

Qj=Tj / ^S Tk

The sum over k includes the entire system. The complexity of the system is defined by the Shannon formula,

C = -K S Qj(lnQj)

The development of the ecosystem can be defined as the rise in average mutual information of the network flow structure. For fik defined as the fractional flow from compartment ‘i’ that will enter compartment ‘k’, the mathematical formulation of ascendancy is:

A = K S S f_ik Q_i ln[f_ik / S f_jk Q_j)].

Ulanowicz suggested that the important factor, K, be equal to the system throughput, T. While T is considered the growth term, A is considered the development term. He further suggests two other terms, E and S. which represent export and dissipation, respectively. Then R, redundancy, is defined as the difference C — (A + E + S) and the overhead as E+R+S. Then ascendancy can be defined in teens of the overhead and the complexity. This allows for a thermodynamical type relationship:

As mathematically represented, the organization of the network contributes most to ascendancy when each compartment in the system sends all inputs, undiminished, to only one other compartment. It can be shown that the organizational term of ascendancy is characterized by specialization, internal cycling and, in the case of focusing on energy flow, minimized dissipation.

In this formalism, Ulanowicz shows that many of the positional parameters accepted by modern ecologists can be represented by a general tendency for a system to increase its ascendancy through throughput (growth) or specialization, internalization, and diversity (development).

Although these extremal principles have been shown, in many cases, to match data, there seems to be inconclusive evidence substantiating vilification in the deterministic sense. Furthermore, mathematical theory has not rigorously proven any of the above mentioned principles. For these reasons it is perhaps more appropriate to revive the original formations. Alternative to mathematical formalism, the verification of these tendencies in nature can be better accommodated through proof by intuition. The groundwork of these principles will most likely not be found in microscopic theory such as the statistical mechanics of thermodynamics. They are better to be stabilized and organized by observation of the whole, by masters in the field. In this modern day, the credit to human intuition sometimes appears insignificant in comparison to the rigid word of mathematics. Perhaps this is not right.

Aside from mathematics, there have been attempts at proof (of ecological extremal principles) by teleology. The ecosystem is considered a living, breathing animal which orchestrates it’s parts to simple unifying principles. These unifying principles are suggested to be one or a combination of the above mentioned extremal principles. Thus, the system moves through time in harmony with itself, globally encompassed. However, this is not the standard format for verifying mathematical principles.

A more appropriate view may be teleological. There seems to be room for the existence of a natural direction. The complexity and beauty of life and ecosystems lend persuasive argument. Darwin mentions this co-adaptation:

Teleological speaking, there need not be an end goal of any type. There are several levels of teleological goals (as defined by Katsenelinboigen, 1992): absolute teleological, characteristic of von Neumann’s turnpike theory, with technologically defined goals independent of initial conditions or resources, semi-relative teleological, with attainable goals defined by the technology and resource, and relative teleological, with unattainable goals set with respect to technology and resource.

Nature evolves via recursion. As coincidentally represented in the simple predator-prey model, the future is not predictable, and therefore effectively indetermined. There may be, however, a teleological direction towards which this recursion is adjusted. As it stands now, this direction has not been completely and clearly determined. Therefore, the preferred navigational tools should be positional Diameters suggested by experts in the field. Rather than utilized in extremal principles, concepts such as exergy, emergy, environs, and ascendancy should be considered the salient positional parameters, capable of defining the predisposition for growth and development in ecosystems. Through continued observation of these parameters, the ecologist’s intuition may become enriched for understanding the teleological direction of evolution. Natural evolution is effectively indeterministic, it should be treated as such.

Quantum Aesthetics

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