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# 45. Step one in defining a species: measuring slight variations:

We have considerably clarified the generation length concept. But convenient as it might now be, it is still not fully scientific. It cannot hold parity with other commodities unless it can be expressed in absolute terms: in seconds, days, years and the like. An underlying question also remains: can biological entities even be construed to behave in a predictable and orderly fashion over biology and ecology?

Our contributions to the above debate are the definitions we offered for both biology and ecology right at the head of this paper. It is now perhaps best to contrast our concise—but above all mathematically precise—definition of ecology with an alternative. In 1927 the English zoologist and animal ecologist Charles Elton tried to offer a definition for ecology by saying:

… It should be pretty clear by now that although the actual species of animals are different in different habitats, the ground plan of every animal communty is much the same. In every community we should find herbivorous and carnivorous and scavenging animals. … It is therefore convenient to have some term to describe the status of an animal in its community, to indicate what it is doing and not merely what it looks like, and the term used is “niche”. Animals have all manner of external factors acting upon them—chemical, physical and biotic—and the “niche” of an animal means its place in the biotic environment, its relationship to its food and enemies. The ecologist should cultivate the habit of looking at animals from this point of view as well as from the ordinary standpoints of appearances, names, affinities, and past history. When an ecologist says “there goes a badger” he should include in his thoughts some definite idea of the animal’s place in the community to which it belongs, just as if he had said “there goes the vicar”. [emphasis in original] (Elton, 1927, pp 63–64).

Elton’s emphasis on doing is encapsulated in the *pdt + mdt* of our own proferred definition and where dt incorporates the activity component. A portion of this doing is maintaining its chemical bond energy of h = ∫p dt. The rest is the energy needed to achieve this, which is δq.In ecological terms, this means that biological entities must always be doing enough work to replenish their stocks of chemical potential and chemical bond energy. Even in hibernation, these are always being claimed with respiration and metabolism being constant and continuous.

We have also summarized Elton’s its relationship to its food and enemies in pdt + mdt. Every biological entity must always do sufficient work to remain viable, and this constantly requires a sacrifice of at least a portion of its own mass and energy, -(du + dh), to the environment. If it becomes prey then the loss is total. Even respiration costs mass, molecules, and energy. It may in a mass and volume of air, but the mass and volume exhausted is surrendered. A biological entity must then replenish both the mass and the energy lost in any work done in respiring and digesting. It must therefore have some specified means—a path or process along with its inexact differentials—that is characteristic of its niche and its overall ecological relationships … and as is suggested by Elton. This is the(p + m)dt whose net effect is the (du + dh) that allows it to remain viable. Thus the niche is the sum of the activities pdt and mdt.

Although Elton’s original niche proposal lacks the mathematical rigour we have tried to give it in our own definition, Rossberg et al make clear why it—along with the emphasis he placed on its relationship to its food and enemies—was widely accepted:

The niche concept has remained of fundamental importance to ecology. It establishes a link between individual-level traits and population dynamics. … The availability of an ecological niche for a species depends, in a great variety of ways, on interactions between the species comprising an ecological community. These interactions, in turn, are affected by the phenotypic traits expressed in the interacting species (Rossberg et al, 2010, pp. 13-24).

One would immediately suppose that such a unifying concept would lead to a thorough-going and rational organization for ecology. But there are of course central difficulties. They are all linked to the etymological fallacies to which ‘niche’ promply gave rise:

However, despite decades of intense research and debate no universal agreement has emerged as to how niches are to be formally represented, and fundamental questions remain unresolved (Rossberg et al, 2010, pp. 13-24).

One major reason for the difficulties with the niche concept is clear from Kant’s definition of geography. It was based on his vision of space and realized in his region concept. Biological organisms are dependent on regions for their habitats. If regions have no set boundaries then it is unlikely that species do. This not only makes niches equally vague, but also throws our quest for an absolute time period for generation length into doubt.

Suppose, as in Figure 44, we want to catalogue the world’s flora and fauna. We make a start at the southernmost tips of Asia and Africa, the world’s two largest continents, and delegate associates to start similarly at the easternmost tips. We all catalogue all biological organisms immediately around us; take one step forwards; and repeat. There is going to be very little difference with each step. But by the time we have traversed the entire length and breadth of each of these continents, we will have embraced as diverse a display of biological forms and ecological niches as the planet has to offer.

Africa and Asia display similar features, caused by temperature, as we move first towards the Equator and then away. A journey along the Tropic of Cancer, latitude 23° 30′ N, will take us across both continents and again show similarities. However, these two journeys will each evidence a huge amount of diversity on each continent, albeit that they are undertaken at equivalent longitudes and latitudes on each.

We can turn to physical geography to resolve our dilemma. Similar kinds of creatures may live in similar environments. Geomorphology offers up the occasional apparently sharply defined lake, canyon, desert, or gorge as a habitat. These might at first appear to establish decisive boundaries, but appearances are deceptive. No geomorphological features can be so easily defined. Even the largest lakes are soon filled in, their beds being ‘deflated’, or soil-eroded, by winds:

As recently as two decades ago, the erosive action of wind in the desert environment was accorded a limited role, and desert landscapes were seen to be largely inactive under present climatic regimes.

… in the last two decades the power of wind to erode desert landscapes has been reassesed. It has been shown to have considerable significance in moulding rock surfaces to produce yardangs, in causing relief inversion, in excavating depressions, and in causing dust storms and deflation. …

… if lake bed remnants of a known age give a clear picture of the former height of a lake floor, then it is possible to calculate the rate of lake floor deflation. Alternatively, dated archaeological remnants, including buried corpses, may become exposed by wind excavation. The presence of aeolian dust-derived sediments in ocean cores may enable the reconstruction of long-term deflation histories from the neighbouring desert source area (Goudie, 1995, p. 157).

Darwin’s natural selection is bedevilled by his phrase “slight variations”. It stands at the heart of his theory. But it is subject to a number of etymological variants. As the power of the wind makes clear, regions blend smoothly and imperceptibly into each other … especially if we are only considering one step at a time. Biological forms and the ecological interactions they give rise to will therefore also have no sharply defined boundaries. As with niche, this is entirely because both slight variations and natural selection lack etymological specificity:

Alfred Russel Wallace, the cofounder of evolutionary theory, was struck by “the utter inability of many intelligent persons” to understand what he and Darwin meant by natural selection and suggested they substitute Spencer’s phrase. When Darwin obliged him by using “survival of the fittest” in later editions of the Origin, readers were still confused; everyone seemed to have his own interpretation of what was meant by “the fittest”.

However, the phrase caught the public’s imagination and became completely associated with Darwin. Critics said it was a meaningless tautology—a proposition that simply repeats itself. Since the fit are the individuals who survive, they argued, wasn’t it another way of saying “survival of the survivors”? (Milner, 1990, p. 424).

The Collins Dictionary of Biology makes this etymological circularity, and the subsequent emptiness of reference, supremely clear:

natural selection, n. the mechanism, proposed by Charles Darwin, by which gradual evolutionary change takes place. Organisms which are better adapted to the environment in which they live produce more viable young, so increasing their proportion in the population and thus being ‘selected’. Such a mechanism depends on the variability of individuals within the population. Such variability arises through mutation, the beneficial mutants being preserved by natural selection (Hale & Margham, 1988).

So … if we read this definition and then want further clarity, it refers us straight back to its own beginning!

Even though natural selection is subject to a wide variety of interpretations, Figure 45 tries to realize the doctrine of the “slight variations” that are the basis of Darwin’s theory. There are three human communities. They are at the same latitude and longitude. They each face similar environments, including a number of steps. Those facing the uppermost community—the original genetic one—are flat. By some unspecified process, two further communities are extracted and placed in equivalent environments. One removed community now spends its time going down then up, foraging and farming in the valley at the bottom, but living on the top steps. The third community does the opposite. We assume that the hills, valleys, and plains are otherwise identical.

If Darwin is correct, then by a process of Darwinian fitness, these three subtly different environments will very gradually lead to differentiation. The proposal is that “survival of the fittest” will eventually lead to disjoint species, albeit that the precise mechanism remains under dispute (Spencer, 1864). Whether the process is micro-, macro-, genetic, phenotypic, convergent, divergent, molecular, chromosomal, the giving of precedence to organic mutability, biological, ecological, environmental, or any other, natural selection is the key mechanism that Darwin points to for his Darwinian evolution. Our biological potential must now not only define its own boundaries plus its objects of study, but it must meet such challenges and provide accurate measures for resolving such disputes or it has little or no utility.

The Cartesian coordinate system that we have so far used to measure biological potential has helped in the derivation of many physical laws. We do not seek to abandon it entirely, but it is less useful in the kind of situation we are currently facing: that of boundary problems involving flows. We have an as yet unresolved issue about periodicity and length. Such boundary matters are often best resolved with alternative coordinate systems. Since a niche involves the inexact differentials of work and heat, δw and δq, specifying the allowed set or the variability of paths referred to in our third law of biology, the law of diversity, is problematic. Our problem is that an allowed set must be specified in terms general enough to permit such variability … yet it must also still set the limits characteristic of that species.

A first and important determinant of any species or population is established by the rotation biological potential makes about a central point. By the Biot-Savart law, any activity over a generation length Δt immediately points to some proposed centre of generation. That implied curvature, and rate of curvature, in its turn immediately implies a biological cycle of given generation length, T.

A generation is a revolution in our biological potential. We certainly want to know not just that a revolution has occurred, but also how long it has taken, and how masses, energies and numbers have interacted across our line segment of molecules for there are likely to be significant temporal differences to contend with over different circuits, never mind the other kinds of variations implied in Law 3 of diversity.

We can make our problem clearer by examining one closely associated family of organisms. There are over 2500 different species of mosquito (McCafferty and Provonsha, 1983). The precise adult life span depends upon sex, humidity, temperature, the time of year born and a variety of other factors. The males live a very short time, averaging around a week. The females live somewhat longer, generally about a month. The mosquito life cycle ranges from four days to a month, depending upon species and environmental factors. The major determinant of generation length is the development period for the juvenile, which decreases as length of the season decreases—and thus with increases in latitude (Masaki, 1967). A representative mean generation length is therefore about T = 20 days.

There could hardly be a bigger contrast with a mosquito than a blue whale. The female is sexually mature at between five and eight years old. The females then give birth to a single calf every two to three years over a lifetime of about sixty years. The males mature in just under five years and are slightly smaller than the females. In their report for the National Marine Fisheries Service, Taylor et al estimated the mean generation length of blue whales as T = 31 years (Taylor et al, 2007).

But not only can generation length vary between species, it can vary within species. We measured generation times for Brassica rapa of 44, 35, and 28 days respectively, with that for the equilibrium age distribution being T = 36 days. There were also, of course, associated changes in mass, number and energy. It is obviously very important to take such differences into account when discussing fitness and natural selection, otherwise no theory seeking to explain the differences between 2500 species of dipterans, or how one can transform into another, can give an accurate accounting:

Evolutionary biologists have long been aware of that pitfall, and many have contributed to making the concept useful. Fitness, it turns out, is a relative term. Organisms that are the “fittest” in one environment, may be completely unsuccessful in another. Or they may be extremely successful for millions of years—as the dinosaurs were—only to be suddenly wiped out when the conditions change.

In populational terms, fitness simply means reproductive success. The race does not always go to the strong or swift, but to those who manage, by whatever means, to produce the largest number of offspring. [Emphasis by this author] (Milner, 1990, p. 424).

We must obviously concern ourselves with both absolute and relative measures for generation length and biological potential. Since we must define a boundary for a generation; and since that boundary does not succumb easily to Cartesian methods; then hunting for a more suitable system of measure must be our first order of business for we must be able to embrace all the different paths and processes permissible to any one population.

Figure 46 fits our purposes. It shows a different method for expressing the locations of points in space (Hughes-Hallett, 2002; Stewart, 2003). It was originally introduced by Newton (Stewart, 2003, p,675). The point P is on a Pythagoras triangle formed from the x- and y- coordinates, which are the legs. Since r2 = x2 + y2, then the distance to P can always be expressed as r = √(x2 + y2).

What is of considerably greater significance is that we can express any differences in x and y as an angle, because y/x is the tangent. If either one changes, then the angle is likely to change. In other words, since our gradient function for biological potential has three variables—for it is derived from our function φ(n, q, w)—then we can express the angle that creates the looping of the generations in terms of the ratio between any two selected variables. Thus if, for example, we make our x- and y- axes n and q respectively—the numbers in the population, and the moles of components per each entity—then a return to a specific ratio between n and q will tell us when we have circumscribed a generation irrespective of the absolute values. This is so irrespective of biological form or of ecological niche occupied.

One generation is one revolution of our biological potential. This occurs when we return to the same value for θ no matter what we measure absolutely in terms of minutes, days or years. As in our experiment with Brassica rapa, we will also know the mass and energy biological potential deploys in creating a group of progeny, over that given time span, and from an initial set of progenitors, no matter what the scale of the increases and decreases exhibited.

The clear advantage of measuring generations with polar coordinates is that the ratios only need to repeat. Consistent with our desire to allow the sizes of our circuits—i.e. generation lengths and their associated parameters—to vary, the specific numerical values can now freely differ. We need only consider ratios. Furthermore, we can always switch axes—which is to switch bases and ratios—and then observe any differences. However we choose to measure it, the rotation that marks the generations will occur over a given length of time, and those time spans can also now freely differ. We are now set free to observe developments, and the relative changes, in all our variables over an absolute time span. The changes in our partial differential equations have acquired changes in angles.

As in Figure 47, these 2-dimensional polar coordinates now allow us to use a 3-dimensional cylindrical coordinate system in which we superimpose our z-axis, from the Cartesian system, over the above polar one. So if, for example, we feel that chemical configuration is the least likely to change in a given situation; or if we feel it is the most neutral; then we can allocate it to the z-axis and allow changes in the other two to define the period of rotation as we observe how they all interact. Instead of the three lengths (x, y, z) that we previously had, we now have two lengths and an angle, reported as (r, θ, z) … which is two quantities and a rate. But we are free to change the allocations at any time. Whichever one we choose for our central axis, every point in space—which is every movement in biological potential—is now properly recorded. We can again measure both the relative and the absolute.

Using the same methodology, we can also express all movements of biological potential in the spherical coordinate system depicted in Figure 48 using one length and two angles. The distance, ρ, comes from ρ = √(x2 + y2 + z2) as the hypotenuse of the Cartesian system; and the two angles are ratios between two properties, however we choose to select them. This allows complete freedom for observing the changes in all ratios and variables and to understand their behaviour, both relatively and absolutely. We are again free to change the coordinates as we wish, and as according to the enquiry. Every molecule in every possible biological entity, no matter what its niche or physiology, is now at our command, and we can discuss their effects and variations much more scientifically and in terms of both each other and the environment, relatively and absolutely.

Relating the relative to the absolute is all very well, but there are a number of immediate issues. In the first place, if two variables x and y have the same relative values at time t1, then this can only be because one, say x, caught up with y at time t1. But there is no cycle if x does not now slow down and/or y speeds up once x has sped ahead, so y can in its turn catch up with x at time t2. But once y has in its turn caught up and swung ahead, x must again speed up and/or y must slow down, so x can again approach y from behind. We are then back at t1. If generations are truly going to repeat, then such relative movements must all take place at the same absolute time points over T. Anything else is evolution, and the natural selection that causes it must then be found.

The other main issue is graphically represented in Figure 49: the need to detect and measure all variations of all possible kinds. In the first icon we have a given cycle of the generations, involving a given amount of mass and energy over a given population. The circumference around the boundary represents the sum of the components and the amount and rate of processing they undergo over the given generation length, T. The arrowhead signifies the value at which given ratios mark the beginning and ending of the generations.

As in the second icon, let now this population maintain the same overall values for mass and energy, but somehow increase in both its mass and its energy fluxes. This is certainly possible because work and heat are inexact differentials. The total number of chemical components and/or the generation length and/or the mass increases as is indicated in the increase in the circumference, which reflects the net increase in work and energy. More work must be done on this increased number of components and/or entities, which is the force applied, and the work done, all around the boundary.

Then as in the third icon, let the population now undergo some kind of change in state such that the beginning of the generations shifts. There could be changes in germination and/or implantation and/or in feeding, or else in delays or advances in juvenile or adult phases etc. There is some kind of change in the ratios making up the biological potential and so in forces exerted, and microscopic components are rearranged in their net processings to effect this, although the total number of components remains the same.

Then, as in the fourth icon, let further changes in the environment now induce a shrinkage in the overall energy used, which is accompanied by a reduction in either the net generation length or in chemical processing, or in mass or in number or in whatever combination as restores the original boundary size. Thus the work and heat evolved are back to their initial values.

And then finally, let this latest population undertake a set of evolutionary changes in state such that all values for mass, energy and number return to the original. We now need to be able to track all such changes, microscopically and macroscopically, so that questions of whether or not all of this is natural selection, and of whether or not an original population can indeed be restored by means of it, can all be clearly answered biologically and ecologically. That is the explicit challenge posed for the etymology of natural selection.

As biological entities do work and evolve heat, they are in constant dynamical interaction both with each other and with the environment. It is important to track all such interactions, most particularly since they involve mass and energy. These are guided by, and involve, microscopic interactions. The fruitful methods for analysing the components of such systems were first introduced by Gibbs:

… Ten years after the law of mass action was propounded by Guldberg and Waage, Willard Gibbs, Professor of Physics in Yale University, showed how, in a perfectly general manner, free from all hypothetical assumptions as to the molecular condition of the participating substances, all cases of equilibrium could be surveyed and grouped into classes, and how similarities in the behaviour of apparently different kinds of systems, and differences in apparently similar systems, could be explained.

As the basis of his theory of equilibria, Gibbs adopted the laws of thermodynamics, a method of treatment which had first been employed by Horstmann. In deducing the law of equilibrium, Gibbs regarded a system as possessing only three independently variable factors—temperature, pressure, and the concentration of the components of the system—and he enunciated the general theorem now usually known as the Phase Rule, by which he defined the conditions of equilibrium as a relationship between the number of what are called the phases and the components of the system (Findlay, 1904).

All changes in mass and or energy require forces of some kind. By noting all such changes, the system’s energies and behaviours can be tracked and analysed. Gibbs’ phase space allows biological organisms to be measured both microscopically and macroscopically.

Biological entities in a population occupy each others’ neighbourhoods. Their paths of work and heat share the equivalence of our Law 2 of biology and so they are energetically “near to” each other. They enjoy a density of points in an energy landscape as they move under energy. They are topologically equivalent in a given energy volume, all across the generation. The biological potential moves such a neighbourhood of entities, which is a population, in a given direction, as also around a point, as it seeks to complete its orbit of the given ratios and values. The Gibbs phase space is energetic and dynamic and represents all the possible states available to any system of points or bodies in interaction.

Thanks to our systems of measure, every transformation that every biological entity undertakes over a cycle can be properly specified by both its position and its energy as it leaves one point and approaches another. If r is the spatial position and π is the ability to do work on given components through imposing chemical reactions upon them—which is to apply a force—then the complete phase and energy possibilities for a system of bodies circulating around at the behest of biological potential is rx, ry, and rz and πx, πy and πz for each entity. Each possible state for each entity, with respect to each axis, and for the entire population, is then represented by a unique point in this phase space, and as ns or qs or ws—numbers of entities, moles of components, or rates of processing—form appropriate ratios with each other.

Figure 50 shows an ensemble of entities in a given population (Denker, 2011). We have number of entities, n, on the vertical axis, and their moles of components, q, on the horizontal one.

We have a population composed of nine proposed sub-populations or sub-species, each contributing to the same overall equilibrium age distribution population. Each grouping has its given n, each of its entities being composed of a given q̅ moles of molecules each. The nine subsets are all tracking the same equilibrium disribution as indicated, and therefore share certain parameters in common. They thus move centrally around the same given point, share the same directional gradient, follow the same circuits, and are subject to the same overall biological potential. Our unit vectors are also in force. This ensemble of qs and ns interacts and loops to form circuits of the generations. It is presently moving from location A to B. Two of the sub-populations or sets in the ensemble are coloured white to simplify determinations of state.

It is easiest to think, pictorially, of the various quadrants in which events occur. The axes form the boundaries to the quadrants. We place n on the x-axis, and q on the y-axis. Thus the populations are on a journey from A to B and beyond, going successively through Quadrants 1, 2, 3 and 4. Quadrants 1 and 2 are positive for the n-axis with values tending to increase in n in Quadrant 1, while they decrease in n in Quadrant 2. Quadrants 3 and 4 are of course negative in n, tending to become further negative in Quadrant 3, while tending to return to the origin in Quadrant 4. Meanwhile, of course, it is Quadrants 1 and 4 that are positive for q, with q tending to increase in Quadrant 4, while decreasing in Quadrant 1. It is negative in Quadrants 2 and 3, tending to become further negative in 2, while tending to return to the origin, and so increasing, in 3. Our unit vectors are in force, with the n = 0 being n' = 1 biomole for the n-axis; and with q = 0 being q' = the generational average for the molar value of the chemical components for that species.

In general, in the left of Figure 50, we see population numbers increasing from the initial n', with A thus being the generation midpoint in values for n, and so midway between the minimum and the maximum values for entity-numbers in each subpopulation and as n moves from Quadrant 4 to 1. However, this same A is also the maximum for the component-numbers, q̅, composing each entity. We may have an average number of entities for the generation, but they are all of the maximum possible size for that generation.

As the generation continues, the ensemble of subpopulations moves to B where entity numbers are at their maximum … but the entities within the subpopulations have decreased in size. They are all at the midway point for the amount of substance held per entity. Component numbers decline all through Quadrant 1 heading towards the Quadrant 2 minimum. We shall assume that, at A, reproduction is underway. There are increasing numbers of ever smaller progeny appearing (n state positive, increasing), thus reducing the overall average individual mass within the population (q state positive, decreasing). Reproduction of course means that DNA is being transmitted from one generation to another within each subpopulation, with each path representing a specified state within the dynamical equilibrium maintained across the generations. The ensemble's potential energy must therefore also be increasing.

As is clear in Figure 50, and as forms the basis of natural selection, this ensemble of entities in those nine subpopulations cannot enjoy exactly the same values for n and q at every point. There are slight variations, of the Darwinian kind, in (a) the numbers of entities held within each subpopulation across the ensemble, and (b) in the moles of components that constitute the entities in each subpopulation as we move from A to B. Some have less, and some have more, of each commodity including transformations in DNA. There are also changes amongst them as the generation proceeds.

As can be seen in the differences in the ensemble at t1, t2, and t3, the overall shape has clearly changed. This is a reflection of the different forces at work. The numbers of entities in each subpopulation, as also the moles of components constituting them—n and q̅—have varied across the subpopulations, and within the ensemble, in response to biological potential. The same would hold for any other choices we made for these axes.

We now have to consider the forces acting on this ensemble, which are displayed on the right of Figure 50. Since our ensemble of entities, in moving from A to B, is in Quadrant 1 and moving anticlockwise towards Quadrant 2, then n is both positive and increasing, q positive but decreasing. We again understand that a more copious number, n, of smaller-sized progeny, q, is being produced. As the system leaves Quadrant 4 and swings towards A, the changes in state towards the n, the number direction, reach the generation average of zero for n, and then head out to the generation maximum at unity or n = 1. Numbers in the subpopulation will continue to increase from Quadrant 1 to 2, but at a steadily slowing rate, as the subpopulations head towards the stationary point at n = 1 … at which point the numbers will begin moving in the opposite direction and declining.

When the ensemble of entities is at B, it will be in a stationary state for the n-axis or direction as the entities prepare to enter Quadrant 3, where n goes into decline. When at B, the ensemble of entities will be in a stationary state for the n-axis or direction as they prepare to enter Quadrant 3 where n goes into decline. Their n will be at a maximum, but their q̅s or components held per entity will be the generation average represented by q = 0. They will also of course hold their maximum—but negative—state-per-unit-time, or velocity, in the numbers of components held. The numbers of components held per entity will have fallen to the generation average, but will then continue in that negative direction, and will steadily decline, but at a decreasing rate, to the generation minimum of -1. Perhaps the progeny, when at B, is comfortably ensconced in seed pods; and perhaps fruits are being eaten by birds and other seed carriers; and/or possibly one hundred million sperms per average male per day are uselessly dissipating, while but a few are using their provisions, and so are busy swimming to fergilize a few eggs, the bulk of which are also being wasted. These are all viable ways of reducing component numbers in the population and consistent with both the behaviour of DNA, and these values.

We can now look at the forces—the impetus impelling these changes. The full behaviour of the impetus force—or the impulse to (a) acquire components; and (b) change in numbers and as encoded in DNA—is as shown in the right of Figure 50. Where the numbers are in Quadrant 1 and moving to Quadrant 2, the numbers-impetus is instead in Quadrant 2 moving towards Quadrant 3 and forcing a decline. At A, the ensemble has its maximum impulse-for-change in its numbers of entities held per second—i.e. to change its velocity in the n-direction—with that impulse then decreasing down the n-impulse-axis as the numbers, which have just now increased above the average, gradually head to the maximum. But we have a deceleration. The force to implement yet further changes in numbers is in its turn declining at the behest of the microscopic components, which is the DNA.

Meanwhile, the ensemble moves increasingly rapidly in the negative q-direction which is also heading from Quadrant 2 to Quadrant 3. There is a steadily declining number of components held per entity. The number of components held was above the vector average when at A, and was already holding the maximum for the population when our analysis began—whereas n was at the average and tending to the maximum. When the subpopulations are at B, the numbers of components held per entity are subject to an impetus that is making those component numbers decline ever more rapidly away from that maximum, which is again the process of producing and provisioning the smaller progeny, perhaps in meiosis, mitosis and so forth, and to transmit to them the DNA required, so that both natural selection and the cycle can continue at each t over T for the generation. When the subpopulations reach B, this impetus to institute changes in numbers is itself at a maximum for its effect, and will begin to decline in its influence as it heads to Quadrant 3. At the boundary between Quadrants 2 and 3, the number of components held will start to increase as a countervailing force is set in train.

Figure 51 now represents the extremely important Liouville theorem about the preservation and conservation of forces around a boundary—and within this “phase space” or “phase volume”—and that therefore helps to govern natural selection. The total for our state-maintained-per-unit-time, which is the system's “kinetic energy”, is the total chemical bond energy, H. The system's total potential-to-change-in-state is always emplaced in its DNA and is its reproductive potential, A, which is its Helmholtz energy and so its potential energy.

We have now defined an actual and a potential—i.e a kinetic and a potential—suitably defined, we then have H = A + H where H is formally called the system’s ‘hamiltonian’. The hamiltonian is in our case overseeing the transfer of energies, on a continuum, between the mechanical and the nonmechanical aspects of the Gibbs energy. This is the working out of the reproductive potential or the Helmholtz energy. It is also the working out of the Franklin factor, K, and the Franklin energy, F, and as according to law four of biology, which is the law of reproduction. The sum of the reproductive potential and the chemical bond energy is course constant for that is how we defined them from Table 1.

The Liouville theorem now states that the hamiltonian, H, will remain constant for the system. In his The Principles of Statistical Mechanics, Richard Tolman neatly presents it as:

(∂ρ/∂t)q,n = -Σ(∂ρ/∂qi ∂H/∂ni - ∂ρ/∂ni ∂H/∂qi)

where ρ is the density of the distribution of our given subpopulations throughout the phase space; q and n are, respectively, the numbers of moles of components and numbers of entities; H is the sum of the kinetic and potential energies for a dynamical system, or the actual bonds plus the potential as encoded in DNA for biology—is the hamiltonian (or sum of chemical bond energy and reproductive potential— H and A—for the system; and the summation on the right is taken over all the i bodies in the ensemble. Gibbs showed, in his analysis of phase systems that this eventually reduces to:

… the simple result dρ/dt = 0, when we consider the rate of change of density in the neighbourhood of any selected moving phase point instead of in the neighbourhood of a fixed point in the phase space. This form of expression may be called, in accordance with Gibbs, the principle of the conservation of density in phase. [italics by original author] (Tolman, 1938, p. 50).

Since, by the Liouville theorem, the net rate of change of the density of points in the neighbourhood of some given system or ensemble acting under energy has a constancy in behaviour, then as evidenced in our Brassica rapa experiment, no matter how the ensemble or system changes or develops, the total of the energy area or energy volume held within a given boundary—which is the sum of the two halves in Figure 51—is preserved as the system shuffles energies within its ensembles. If one property increases then some other property diminishes elsewhere and commensurately … and as we saw with B. rapa; as Richards and Waloff saw with Chorthippus brunneus; and as Pakenham and Lewington & Parker reported with the bristlecone pines Pinus aristata and P. longaeva. The total energy in this system—which is an equilibrium age distribution population—is simply the sum of (a) the chemical bond energy expressed; and (b) the reproductive potential as we have earlier defined it.

As a given system of energy switches between its manifestations of its expressed and its potential and other energies, the Liouville theorem insists that overall phase volumes or energy totals will be preserved by the microscopic constituents of the ensemble or system because, by the first law of thermodynamics, all energy is equivalent. And as Mayer also showed—when he first discovered the energy concept—biological entities survive entirely by converting to and from mechanical energy and incorporating that into themselves as both the constant pressure and constant volume expansions, or as the mechanical and the nonmechanical forms of chemical energy.

Figure 51 contains suitable plots of the states held against the impulses-to-attain-states encoded in DNA for the independent q- and n-coordinates. The ensemble’s behaviour is once again the sum of the values in the two halves, one for q, the other for n. They together depict a total area—a phase space—shared by all points or entities within a given neighbourhood of energy as members of an ensemble, and as participants in the suite of coordinated energy interactions that is the equilibrium age distribution population.

The ensemble of points stating the behaviour at t1 for q at top left in Figure 51 are in Quadrant 4 and moving into Quadrant 3. They are relatively open. Meanwhile, those at the top on the right for the same t1 for n, which are in Quadrant 1 and moving down into Quadrant 4. They also are the more closed. These differences in quadrants and in states of open and closed reflect their different effects on the system. At t2 the qs are almost completely closed, whilst the ns have opened. The ns become even more open at t3, while the qs have opened from their previous near closure … but our white marker subpopulations are now on the other side.

It is now important to remember that our line segment of molecules, and all forces arising from it, are vectors. We have accelerations and decelerations in play. Sometimes forces tend to increase the entities’ components, numbers, and energy … and sometimes forces tend to decrease these. They have a sense and a direction. Figure 52 illustrates the vector conventions for these forces. They still follow the right-hand rule. The unit normal protrudes from the positive side of the surface, the other side then being negative.

If a force now acts so that we could walk in its direction of action, which is then anticlockwise, and keep an increase from the unit normal on our left, then the force is positively oriented around the boundary. If these conventions do not hold, then the force is acting negatively. In Quadrant 1 we will see objects and or forces above the zero-average point and tending to increase towards and beyond the vector normal, while in Quadrant 2 we will see objects still positive and above the zero-average point but tending to decrease towards that point or origin. In Quadrant 3, values will tend to be below the average or zero and tending to move further in that direction, returning in Quadrant 4. Whether they are on our left-or right-hand side depends upon both our orientation to the normal, and the direction in which the force is working as we move around the boundary.

The area formed by the subpopulations in q are closed at t2 because they are busy reversing their direction. Component numbers were maximum positive and an impetus exists to take them to maximum negative. Meanwhile, the subpopulation values in n have already changed direction. The impetus is still to increase in n, but it is reversing and decreasing as the rate of change is slowing. The white markers therefore work in one direction at t1 … only to reverse it for t2 and t3 as the ensemble moves from Quadrant 1 to Quadrant 4. At t1 the white marker for the qs at A is at top left. These are busy moving from Quadrant 4 to Quadrant 3. Its neighbour marker is down anti-clockwise around the boundary, meaning that both the state and the impulse to do work on the line segment of molecules are tending to increase in that negative q-direction over the ensemble … which is to reproduce and to produce smaller entities. The direction of the force or impetus around the boundary is therefore anti-clockwise, which is always counted as positive for force.

Something is causing these bodies to accelerate increasingly negatively with time and state, although the actual distance travelled or change in state undertaken in the q-direction is currently relatively small. It could be that many seeds—but all of the same sizes—are being produced. The net energy in this neighbourhood, due to the force acting on the q-axis and on its entities, is therefore tending to increase the q-activity, which is to encourage a partitioning and a net decrease in component size. As can be seen at A in Figure 50, the bodies are at their maximum q-position so they can go no further rightwards, since the moles of components held per entity are at a maximum and are beginning to reverse. We have our progenitors, and we now need our progeny. The entities are thus under maximum acceleration in the n-direction, and towards B. However, although also under acceleration in q, they are busy reversing rather than undertaking any changes in state in q, so they move but little or even none in the q-direction, which is components held per each unit of time. The size disparities are small. Thus the number of components held per entity is almost stationary from one t to the next. And although a deceleration in n is now being initiated, it also as yet has little effect, so the numbers are still on the increase and at a rapid rate … although the imperative to further increase is in decline. Thus the forces in q are large—meiosis and mitosis have possibly begun—but the actual q-distances or changes in state are small, so the ensemble is more spread out along the q-axis and favours that direction, as the forces on n go only slowly into reverse. This large movement in one direction but small movement in another produces large numbers of same-sized entities and also accounts for the small area of that ensemble in q at t2 in Figure 51.

The white marker body for the ns in the right-hand figure of Figure 51 and at A is at the top right of that ensemble at t1. The body next to it is now in the clockwise direction as the figure moves from Quadrant 1 to Quadrant 4. This force is acting oppositely around the boundary, and making its properties diminish. We have one anticlockwise force upon the qs; and another clockwise one in n. This entire area of energy in n must therefore be subtracted from the area for q, because n is in the net decelerative on the ensemble, while q is accelerative. The former is striving to change, while the latter is striving to prevent a change.

At time t2 the combination of forces and positions over on the left on q makes the net energies virtually zero. The acceleration has almost reached its maximum and is beginning to diminish—although the velocity or state energy per unit time itself continues to increase only gradually as the acceleration gradually reverses until its maximum at q = 0. The force in the negative direction along the q-axis may have diminished, but the velocity is great, and so there is far more movement in the q-direction per each unit of time. The generation average of components held per entity per unit time has been attained and those numbers are currently diminishing at the fastest rate, and the change will now begin to slow.

Since the net contribution of active force for the qs is virtually zero, then the system’s net energies are almost wholly represented by t2 in n—which is (approximately) the “Liouville constant” or “Liouville number” for the system. That number represents its total phase volume which is its total in energies and energy interactions and is a specified volume.

When we look at the white marker bodies for n at t2, they are at top left, but we must this time move anticlockwise. The net contribution of force in n has switched from negative at t1 to positive at t2. And then at t3, the ns have opened out considerably and still have a positive sense. However, we must now subtract the qs at t3, which is at B, because q = 0, which may be the average for the generation, but it is also the maximum in the decelerative force. A reversal of the impetus, or the impulse, upon the number of moles per entity in the system now sets in. The number of components per entity continues to decrease but at a decreasing rate. This area must now be subtracted from that for n at t3. It is reversing. When the two are combined the total phase volume has remained the same.

The net result of these interacting forces is that any expansion in one direction, along any coordinate, is immediately matched by a balancing shrinkage on its conjugate axis. The Liouville theorem states that the phase distributions of all entities and molecules will have a constant phase area and/or phase volume. The density of points acting under energy in any given location or neighbourhood will therefore always be constant.

Figure 53 states the net consequences of the interaction of phase spaces, the Liouville constant, and the hamiltonian of energy (Kennett, 2007). The phase space is free to change its shape—and will change its shape—and so will gradually and systematically explore every permutation available to it. Thus the volume remains the same as the phase shape morphs from the configuration it has on the left side of the arrow to the one upon the right. The boundary then encloses a region or volume of constant energy … and is ready for our planimeter.

The ensembles that establish a given phase volume are completely free to explore the various trajectories made available to them within a given phase space or volume, and as is consistent with the boundary defined by that space at any given time, and with the net quantity of energy made available to and within that volume, and as the entities constituting those subpopulations exchange energies between the actual and the potential—which is to exploit their DNA—and also vary their numbers at each t, along with the moles of components available in each neighbourhood of the phase volume, and therefore as compose each entity.

There is of course a difficulty. It is never possible to exactly determine the integral curves—i.e. the forces—active in any vector field (Swetz, 1994). But by Birkhoff’s ergodic theorem a “time average” can be determined. We simply follow any one molecule or particle, calculate its energies, and then average those energies over the time period concerned. This time average is then equal to a “space average” we can also compute by instead taking a given moment, and measuring the energies of all the molecules at that given time throughout the given space. We can then take that alternative average to represent the system. And since the time average equals the space average, the present average value of any system can be used to predict its future average value, which is the value of the energy neighbourhood concerned.

We presented an ensemble of nine subpopulations, all within a given population … and as if all nine existed contemporaneously. This was to make the Liouville theorem and its implications for biology easier to present. However, by the Birkhoff ergodic theorem, those nine subpopulations could also and in fact have been a single population but that pursued nine different trajectories over nine different generations … or any variant thereof … and for any number of subpopulations … and any number of generations.

The Liouville theorem is a completely general statement. It establishes the conditions for the second law of thermodynamics; for the Heisenberg uncertainty principle; and for many other foundational principles of physics and of natural laws involving force and energy. All orbiting systems—including biological entities acting under biological potential—abide by this theorem. It therefore establishes the boundaries within which natural selection—which is measured in watts as biological potential and is dS or µ—makes its circuits round and round the generations. The theorem and this potential together establish the length and the nature of the generations as the entities contained within it do work and emit heat in their inexact differentials, as well as establishing the neighbourhoods and the permissible changes in state for any and all axes, and in their Gibbs, reproductive, and Franklin energies. Those Gibbs, reproductive, and Franklin energies are held in common by all the entities that share that common boundary, and thus help to define them all for they hold an equilibrium age distribution in common. We can measure all their changes in state, and also all the forces that cause their mutual changes in state, and the energies they exchange amongst themselves in common with their environment, and both relatively and absolutely.

Natural selection is now the phase volume that all those entities and subpopulations share as they do work and emit heat in their inexact differentials through their ecological transactions and with their DNA which at one moment expresses itself phenotypically, and then potentiates genotypically, but all the time possessing given and measurable quantities of energy. We can now observe—and measure—all variations which is the totality of biological potential and natural selection. We have a boundary of constant energy and constant force. This is natural selection: ‘a power incessantly ready for action’.We have at last found a way to measure it.