21. The first maxim of ecology: the maxim of dissipation

James Maxwell used a “Gaussian surface”—an entirely imaginary construct—to study electromagnetic effects. Its unique properties are ideally suited to our own analytical purposes. It allows us to use one of the four fundamental theorems of the vector calculus. It handles volumes by relating them to areas. It will therefore assist us in establishing sizes and boundaries to enclose biological systems.

The most important aspect of a Gaussian surface is simply that a flux—howsoever defined—moves through a surface that slices across it, and which has an area. The simplest of all assumptions is that the area, at the point of measurement, is flat; and that the flux is constant over that infinitesimal area (see inter alia Fleisch, 2008). The calculus becomes more advanced when these assumptions are probed, but the essential principles remain the same. The Gaussian surface is rapidly adapted. It is virtually the definition of calculus that it is always possible to divide an area up into infinitesimally small squares, through each of which the flux passes. Those small squares can then be treated as flat, with the flux passing through them being both uniform and orthogonal for that specific infinitesimal sector of the surface. This is then about limits … and integration and differentiation follow. It is sufficient to know that vector calculus can get as complicated as we need, and that it can overcome all potential objections.

Any surface, imaginary or otherwise, that completely encloses a three-dimensional space can be considered a Gaussian surface. It is simply one in which field or energy lines associated with our object of study—how ever we choose to define them—both enter and leave.

Reproduction is biology’s signature event. If our programme is to succeed, then we must certainly describe it with powerful and symbolic mathematical language. But we immediately begin with an anomaly.

Given our above definitions of intrinsic and extrinsic, reproduction is intrinsic rather than extrinsic. A more formal definition must therefore await the derivation of our four laws of biology. Since, however, reproduction has an implied tension with—for it acts counter to—the extrinsic degradation and dissipation imposed by the second law of thermodynamics, we must acknowledge and anticipate it as we derive our maxims.

  1. Maxim 1: Proviso 1

Since the first two laws of thermodynamics are universal and so are applicable to all possible systems, they establish the parameters within which biological organisms must operate. The first announces the existence, equivalence, and fundamental properties of energy. All transports of energy—including the biological—must begin and end on some material source possessing mass. But by the second law, if a given biological organism maintains a specified quantity of chemical components at a given rate, then it cannot do so indefinitely. It will eventually surrender its biological nature to the surrounding environment and dissipate heat. By the second law, no organized centres of work and energy can last indefinitely. We can best express this dissipative requirement by following the mathematical formalism first proposed by Rudolf Clausius. We therefore have:

  1. dm < 0

where the moles of the amount of substance of the various chemical components maintained by that entity is reckoned as m kilogrammes per second. This proviso implies that any and all mechanical or other work and/or energetic activities of whatever kind being engaged in by any specified entity will eventually come to an end and it will degrade.

  1. Maxim 1: Proviso 2

As for the tension with reproduction, the famous Louis Pasteur experiment of 1859, refuting spontaneous generation, was a seminal moment for biology. In 1864 he correctly assessed its significance by saying: “Never will the doctrine of spontaneous generation recover from the mortal blow struck by this simple experiment” (Dobell, 1960).

Figure 17: Towards a Biological Flux
 Towards a Biological Flux

The bar magnets in Figure 17 may not seem immediately relevant to Pasteur’s discovery, but there are distinct deep structure similarities. Magnetic field lines are analysed using Gauss’ law and a Gaussian surface. These again link an area to a volume through the surfaces surrounding that volume.

By Gauss’ theorem, which is the divergence theorem, magnetic field lines form circular loops. The field strength at any one set distance r is invariant all about that magnet. The relevant field lines enter a magnet and emerge undiminished the other side. Since everything that enters also leaves eternally, all such lines loop invariantly and continuously.

The differential form of Gauss’ law, as developed by James Maxwell, describes the above behaviour as B = 0 where B is the magnetic field strength, and nabla or del is the differential vector operator. The del dot relation states that we will never find a rate of change in magnetic flux along a field line while looping about it at any given distance r from any magnet. The amount of magnetic flux entering any unit volume, at any specified distance, is identical to that leaving that same volume. Magnetic activity exhibits the same magnitudes at both the entry and the exit surfaces of that ΔV. If not so, then flux would be either emerging out of, or else be entering into, some specified volume element without doing exactly the same at the other side. We would in other words eventually find a point magnetic source with some field line terminus … which has never yet been done. Truly independent sources for magnetic energy simply do not exist. We will never find a magnetic north-pole isolated from a south one. Thus the divergence in magnetic flux is always zero.

Let us now reconsider the Pasteur experiment. Thanks to it, spontaneous generation became moribund and was replaced by William Harvey’s dictum omnia omnino animalia, etiam viviparam atque hominem adeo ipsum, ex ovo progini or “almost all animals, even those which bring forth their young alive, and man himself, are produced from eggs” (Encyclopaedia Britannica, 1875, 1902). Truly independently sourced biological organisms—i.e. arising entirely from some terminus in inorganic matter—do not exist. All biological entities diverge—in this strict mathematical sense—from others like themselves. Every biological entity has a progenitor. No independent point-biological sources—i.e. without a progenitor—exist.

Since Brassica rapa is semelparous, then in spite of the manifold activities the entities engage in, none survives for more than one generation. Since all biological entities, semelparous or not, must ultimately follow our Proviso 1 and degrade, then any population must at some time be—in vector terminology—a net source for biological matter. That is to say, biological matter emerges out into the environment. Entities are lost from the population at a faster rate than they can grow or be reproduced. This is a net, positive, divergence in biological material. It results in an increasing outwards flux. For each moment or “volume element” there is therefore a net loss of biological material, and a net dissipation, again as according to our Proviso 1 of ∫dm < 0.

Figure 18: Divergences in Biological Matter
 Divergences in Biological Matter

As in the left half of Figure 18, if we follow a divergence backwards, against itself, in towards its centre, for long enough, we must eventually find its point of emanation: the ultimate material substances engaging in those biological acts of dissipation, and that are emitting that ongoing flux. But this also means that if we observe the divergence for long enough, and this dissipation into the environment continues, we will eventually observe the total annihilation of the entire population as it becomes extinct.

A total annihilation of biological entities and materials is all very well, but the Brassica rapa population in our experiment demonstrated a continuing equilibrium age distribution. Table 1 informs us that before the population’s total number of chemical components maintained does in actuality become zero, a net vector sink instead sets in, which is seen on the right of Figure 18. A complementary phase of the net appearance of biological matter into the population manifests itself. Those processes pull mass into the population, from the environment. This is growth and reproduction. Biological matter therefore appears faster than it leaves and we have a net inwards flux. Our previous net positive divergence has been followed by a net negative one, or convergence.

When we now follow this negative divergence into itself, in search of an ultimate sink or point of emanation, although it gets smaller and smaller and we zero in on points that could be the source of the relevant biological matter. But we instead find that the biological matter we are following eventually does the opposite. It instead begins to diverge and we see our net outwards flux being repeated. We will now oscillate from sink-like to source-like behaviour and back again … but some biological matter is always and continuously present for we never reach a terminus. There is no true beginning to biological matter. A change in the direction of the divergence is always imminent.

If we therefore consider our Brassica rapa equilibrium age distribution population as a whole, then positive and negative divergences ultimately balance out. The population is consistently maintained. When divergence is positive, such that valuable biochemical components are being lost overall, then it is eventually going to turn negative so that there is a net gain; and conversely. We have described an oscillation around some as yet undetermined mean. We must of course eventually determine both that mean, and its cause. But recollecting that any dy/dx is not in fact a ratio of two infinitesimals, but rather the limits of the ratios of given finite quantities, both of which tend to zero; then courtesy of that standardized “abuse”, so-called, of calculus notation (Stewart, 2007, Section 4.4); then rather than the B = 0 of magnetism, we now in fact have Proviso 2:

  1. M → 0.

Proviso 2 says that the divergence in biological matter always tends to zero. Whenever it is positive or increasing, it will eventually turn negative and decrease; and again conversely. First we have a net inwards flux of biological entities and matter, then a net outwards one. The divergence is zero at least twice over the generation, at which points the flux is solenoidal or like that of a magnet. But although constantly oscillating, it averages zero overall. And … an overall divergence of zero has the same immediate implication as in magnetism: spontaneous generation never arises. Before we can stumble on an independent point biological source in inorganic or non-biological matter, we loop seamlessly back to a set of preexisting biological organisms in which we see a reversal of activities, and then away again to another set. The net divergence is zero because the divergence in the flux of adults have acted as progenitors and left behind them the convergence in the flux of progeny. And since these divergences, fluxes, and changes in divergence are entirely measurable, then our use of vector calculus has already given us a most important—and easily validated—result.

We still have to explain the continuity in chemical components across the generations, but this is nevertheless a considerable advance and should give us some faith in our continuing search for vectors and their differential equations.

  1. Maxim 1: Proviso 3

James Maxwell understood that divergence is a flux density. He saw that divergence is not the total flux, but rather the flux per unit volume. As a measure of density it is conceptually and importantly distinct from the total flux.

Since it is important to keep these ideas of the flux and its divergence distinct, it is very important to be clear what “volume” is, in this context. We have already had an etymological wrangle with volume in thermodynamics, and it is therefore not wise to make the immediate assumption that we know how it should be used in this flux and Gaussian context.

Biological populations do not seem to “have” a volume in the mathematically extensive sense where F({xi}, {Xj}) = ΣjXj(∂F/∂Xj). The volumes of individual entities do not sum to produce a volume in the way that spatial volumes do, which is perhaps what Harte et al commented on (Harte et al, 2008). However … the same is strictly true for electric and magnetic charges which also do not have a volume in that sense. This is again a matter of how terms are defined.

Figure 19: An Electric Field
 An Electric Field

Volume seen simply as a three-dimensional cube is in this case misleading. Figure 19 clarifies volume as conceived in the Maxwell electrostatic equation. The electric field—very different from the magnetic one we saw earlier—is radial. It does not loop around. It instead emerges from a definite point, and carries a definite value. At any given distance r the field strength has decreased by a proportion based on 1/r2. That is to say: (a) the total amount of charge contained in any field is finite; and (b) the strength of the electric field tends to zero as the distance tends to infinity. Since the field is radial, the ‘fall off’ in field strength is exactly compensated for, all around it, by this reduction by 1/r2. But as this field weakens with increasing distance, it does not do so any more strongly in any one location, at any set distance r, than at any other. The fall off is everywhere the same. Therefore, the divergence is zero (almost!) everywhere and is always proportional to the charge density being experienced at all other locations also at r. No points at any given r differ from any others at that distance all about the charge.

Evaluating field strength at the origin itself is a little less simple because = 0, and division by zero is not allowed. Evaluating a charge’s divergence at its point-source therefore requires the formal calculus-based definition of divergence based on limits. Divergence is then a limit over the given infinitesimal volume element ∆V, as ∆V shrinks to zero (Fleisch, 2008). This reduces to the average charge density within ∆V, which is then simply the charge density at the proposed origin, which is simply where the proposed point-charge is located. Therefore, there is a nonzero net flux through an infinitesimally small surface immediately surrounding the point charge at its origin. If a net flux now emerges through that infinitesimally small surface, we say there is a positive charge located there. If a net flux enters we say it is negative. The electric field lines therefore move directly from positive charges to negative ones, diverging nowhere else. I.e. the net divergence everywhere away from an origin remains zero, with the field strength falling away with distance. Therefore, the flux density in all other locations around a point charge depends directly upon the quantity of charge at that origin, for this now determines the quantity of flux emerging or entering. There is still no distinction amongst any points at r away from the origin and so no divergence anywhere else. The total energy flux at each r depends only upon the charge density at the origin … and so “volume” is now simply a count of point charges.

Since volume is now a straight forward count of the point entities producing the field, we can easily transfer this idea to a biological population and to our proposed biological Gaussian surface. We simply count entities. We will justify our position considerably more formally later, but population count is then biological “volume”. And since divergence is simply the flux density per unit volume; and since the total mass flux of a biological population is the M kilogrammes per second of chemical components its members process; and further since the population number at any time is n; then the divergence in a “biological field” is  = M/n. And since we are presently more interested in the total flux, then we have:

  1. M = nm̅.

We now have three provisos. They give us three pieces of mathematically precise information pertaining to biological populations and their extrinsic relations with the environment. They also take us considerably closer to establishing a boundary and identifying biological inertia.

The proposed linguistic rendition of these three provisos should not obscure their underlying—and powerful—implications, nor the deep structure undergirding their mathematical reality. The verbal and the mathematical propositions say the same thing … but one is distinctly measurable and permits of precise calculations concerning the extrinsic aspects of natural selection and competition and the relations between biological entities and their environment. We have also successfully used Gauss’ theorem, one of the most powerful and successful in the pantheon of physics, and linked a double to a triple integral and related them to biological entities. The three provisos give:

The First Maxim of Ecology: The Maxim of Dissipation

dm < 0; M → 0; M = nm̅

[Darwin’s theory of competition]
(A) Any entity that can lift a weight will be prevented from so doing; and/or (B) can be put to use for the same purpose. (C) No entity can lift a weight indefinitely.