Previous: 43: The natural selection of Brassica rapa  Next: 45: Step one in defining a species: measuring slight variations:
44. A time scale for natural selection
Our vector calculus and the biological potential we have discovered are of great assistance in resolving yet another contentious issue in biology and ecology. This revolves around the fact that natural selection is all very well, but Darwin’s great gift was turning the debate to competition … to variation in n … to how the biological entities within a given population contend with each other as they each strive to leave progeny.
Darwin certainly recognized the issues involved, but could only offer such vague generalities as:
It may metaphorically be said that natural selection is daily and hourly scrutinising, throughout the world, the slightest variations; rejecting those that are bad, preserving and adding up all that are good; silently and insensibly working, whenever and wherever opportunity offers, at the improvement of each organic being in relation to its organic and inorganic conditions of life. We see nothing of these slow changes in progress, until the hand of time has marked the lapse of ages, and then so imperfect is our view into longpast geological ages, that we see only that the forms of life are now different from what they formerly were.
In order that any great amount of modification should be effected in a species, a variety when once formed must again, perhaps after a long interval of time, vary or present individual differences of the same favourable nature as before; and these must be again preserved, and so onwards step by step (Darwin, 1869, p. 9697).
What is of interest here is that there has been no significant advance on the ‘daily’s, ‘hourly’s, ‘slow changes’, ‘hand of time’s, ‘lapse of ages’, ‘longpast geological ages’, or ‘onwards step by step’s with which Darwin peppered his writings.
We have here yet another example of biology and ecology lacking clarity in a truly core variable. Without a clear definition of the time scales applicable to natural selection a dn/dt relating to competition and evolution cannot be computed and cogent scientific discussion cannot take place. Erroneous suppositions cannot be refuted. It is therefore critical to determine the appropriate time period.
In his classic paper The Population Consequences of Life History Phenomena, Lamont Cole begins auspiciously by saying …:
The neglect of the analytical methods by biologists may be attributed in part to the tendency of writers in this field to concentrate on the analysis of human populations and in part to skepticism about the mathematical methods of analysis (Cole, 1954, p. 103).
… and he then establishes his framework by saying:
If it is to survive, every species must possess reproductive capacities sufficient to replace the existing species population by the time this population has disappeared. It is obvious that the ability of the ancestors of existing species to replace themselves has been sufficient to overcome all environmental exigencies which have been encountered and, therefore, that the physiological, morphological, and behavioral adaptations that enable offspring to be produced and to survive in sufficient numbers to insure the persistence of a species are of fundamental ecological interest (Cole, 1954, p. 104).
Cole then goes on to discuss many different kinds of life histories and tables and strategies … but he nowhere states exactly what a generation length is, and nor does he define one.
In 1932, Max Kleiber related body size to metabolism in his famous 3:4 body size:metabolism power ratio (Ginzburg and Colyvan, 2004). By the Kleiber allometry if a first animal is 10,000 times bigger than a second, then the first will also have a basal metabolic rate that is 1,000 times larger. In his 1965 Size and Cycle, J. T. Bonner proposed that maturation time had a 1/4 power ratio to animal body size, with an animal 10,000 times bigger taking 10 times as long to mature. In 1974 Tom Fenchel proposed his Fenchel allometry by which the same 1/4 power ratio relates reproductive rate to body size, so that an animal 10,000 times bigger not only takes 10 times longer to mature, but also reproduces at 1/10th the rate. And as another tweak, in 1981 John Damuth proposed that the average numerical density of animals and birds in a natural habitat declines by a 3/4 power ratio based on body size such that if one mammal is 16 times bigger than another, then there will only be 1/8th as many of them per unit area of a given environment.
These allometries are certainly interesting, but:
 rather like the relationship between Kepler’s and Newton’s laws, they offer no explanation for the phenomena they describe and do not discuss rapidities or relative scalings;
 they are subject to great variability;
 they make no mention of the molecules of which the entities are composed; and
 because of (a), they again offer no rigorous definition of the generation length concept.
When molecular biology hit its stride, there was of course an immediate search for the possible evolutionary role played by molecules, such as with Plank and Harvey who offered their Generation Time Statistics of Escherichia coli B Measured by Synchronous Culture Techniques:
Considerable variability exists in the generation times of individual bacteria in a population. The nature of the distribution of these generation times has been the subject of a number of studies. These workers observed the individual organisms directly and for Escherichia coli, for example, unequivocal conclusions as to the form of the generation time distribution could not be drawn. Considerable uncertainty also exists concerning the nature of the correlations between the generation times of related organisms, particularly those between mother and daughter cells (Plank and Harvey, 1979, p. 69).
So in spite of the switch to molecules, there is still no definition of generation length.
Little progress seems to have been made by the time of Martin and Palumbi some fifteen years later, who draw broadly similar conclusions:
The search for factors that influence molecular evolution is made more difficult because many physiological and life history variables are correlated with one another. Such variables include generation time, life span, age at first reproduction, intrinsic rate of population increase, population size, and weightspecific metabolic rate. Traditionally, these variables have been related to a single, easily measurable biological attribute: body size. …
Body size probably does not control the rate of DNA substitution directly but serves as a convenient guidepost for understanding the biological correlates of molecular rate heterogeneity. We attempt to illuminate the underlying causes of the body size effect by separating the influence on evolutionary rate of some of the attributes correlated with body size. We primarily focus on generation time and metabolic rate because the effects of both on mutation have a sound mechanistic basis, have been suggested to play important roles in determining rates of molecular evolution, and represent factors for which large comparative data bases exist (Martin and Palumbi, 1993, pp. 40874091).
But although Martin and Palumbi place the words “generation time” prominently in their title, they conclude that … :
Although the generation time hypothesis has successfully explained some of the variation in molecular rates that have been observed, the above section documents instances in which its predictions fail to be met. Other factors in addition to generation time undoubtedly play important roles in molecular evolution. In particular, metabolic rate has been mechanistically related to the molecular process of mutation. In fact, many exceptions to the generation time hypothesis previously noted can be explained by differences in specific metabolic rate (Martin and Palumbi, 1993, pp. 40874091).
So they also singularly fail to give a definition, molecular or otherwise, for this generation length.
And while Gillooly assures us that:
Generation time is the single most important determinant of the rate of population growth in species and thus, differences in generation time affect numerous ecological processes, including intra and interspecific interactions (e.g. competition, predation). Differences in generation time (time from egg to maturity) among species are sometimes attributed to differences in body mass, as generation time has been shown to scale positively to body mass for diverse assortments of plants and animals and within taxonomic groups of endotherms (Gillooly, 2000, pp.241–251).
he also does not offer us a definition.
The position is perhaps best summarized by Charlesworth who wrote, in Evolution in agestructured populations, that: “Since the concept of generation time is a rather arbitrary one in the context of agestructured populations, several alternative measures have been proposed” (Charlesworth, 1994). To make the point, he then lists several alternates used for a variety of reasons throughout the literature.
There is in fact good reason, on this occasion, why biology and ecology have failed to tightly define a generation length. It would effectively define a species … for which there is also no easy definition. If tight generation lengths of the kind here being alluded to were indeed possible, then Darwin’s “slightest variations”, would in fact be impossible and there would be no evolution. We must therefore create a definition for a species that is rigorous … yet at the same time leaves its boundaries porous so it can be malleable. This is certainly a challenge, but vector calculus makes it possible.
We can take a leaf from Immanuel Kant’s book. He proposed three methods for systematizing knowledge (Elden and Mendieta, 2011; Casirer, 1981). The first was to classify facts according to the type of objects studied. Thus, following Kant, we can say that population biology studies generation lengths, genetics studies genes, herpetology studies snakes, and numismatics studies coins. The second was to classify facts temporally, which is essentially history. Kant’s proposed third method for organizing knowledge was spatially, which was geography, and also anthropology:
I treat geography not with the completeness and philosophical exactitude in each part, which is a matter for physics and natural history, but with the rational curiosity of a traveller who collates his collection of observations and reflects on its design (Quoted in Casirer, 1981, p. 52).
Geography fitted into Kant’s understanding of the world because it was how places—matter—were differentiated. It was to him a discipline that synthesised the findings of other sciences, using its concept of area or space. The human was the synthesizer, and Kant divided geography into six subdisciplines: physical, mathematical, moral, political, commercial, and theological geography. But the essential object geographers study, so Kant declared, is the ‘region’.
Kant’s definition of region is useful to us because it offers no set boundary, yet each region has its distinctive nature. Being spatial, a region cannot be moved. All regions are also different or there would be nothing for geographers to study. A region’s first characteristic is therefore its natural environment: its landform and its climate. Its second is the relation between its physical elemental complex of attributes, and how human beings have related to that complex in the past. Its third is its sociocultural context in so far as these are not easily replaced, or else are the stable structures that await new arrivals or immigrants.
The size and nature of any region thus depends as much upon the geographer’s chosen scope of investigation as it does upon the region itself. On this reading, a lush tropical jungle is just as valid a region as a South American country.
The vector calculus model we have developed now shows an important advantage. It allows us to produce a definition for a generation length that is clear, rigorous and unambiguous … but that still allows for the flexibility and variability that Darwin’s theory for the evolution of species requires. It is defined in terms of our new biological potential, µ.
Figure 43 shows a topographic map with three axes. We have depicted the gradient ∇µ at the point t … which could easily be a given time point in a given generation length T. This µ is of course, our field of biological potential … but the principles may perhaps be easiest understood by considering it as a physical hill.
We can clearly walk and/or run around the hill that we propose on many different trajectories, and at many different speeds … yet complete a circuit each time. On good days we can go faster and tackle some steeper gradients, on bad days we will go more slowly and avoid slopes. We can always determine the direction to the highest point within this field … or on this hill (Nykamp, 2011b, 2011c).
We now switch our mapping back to biological potential and the changing effects of the environment and competition … and we simply have to express these possibilities for gradients slightly more formally. If we now begin from the point t, we can soon determine the direction of steepest ascent. It is the way to restore equilibrium and we therefore try to move in that direction. But there is a force in operation—say an inexact differential and/or the environment—that sweeps us slightly off our intended course. We therefore end up in the infinitesimally nearby location t + dt. However, the gradient ∇µ(t + dt) still exists and it still points to the same highest point … and we can keep trying to move up towards it. We can now move or be swept all around T, and so over every t, and always be seeking to head towards this same point of maximum increase, for it is the expression of our response through a potential, ∇.
We now express the situation using the x, y, and zaxes. Every axis has its scale of measure which we must somehow determine. The accompanying infinitesimal change in the field, dµ, when we move from t to dt is the sum of all the separate changes on each axis and in each direction. It is:
dµ = (∂µ/∂x) dx + (∂µ/∂y) dy + (∂µ/∂z) dz.
But although we have yet to determine their values, we can express these distances through standards of measure applicable to each axis. There is some distance or representation, whatever it may be, that stands for the distance from zero to one on each axis. These “unit vector normals” are the weighted averages we mentioned in presenting our population equation and are generally represented by î, ĵ, and k̂ for the x, y and z axes respectively.
Assuming such units of measure for each axis now exist, we can reexpress the infinitesimal distance we travel at any time as:
dt = îdx + ĵdy + k̂dz.
And with this expression for distance we can express the change dµ using our gradient operator as:
∇µ = î(∂µ/∂x) + ĵ(∂µ/∂y) + k̂(∂µ/∂z).
This is of course now more felicitously written with the del operator as:
dµ = ∇µ • dt.
We now have a solid “directional derivative”, as it is called (Nykamp, 2011b, 2011c). Its rate of change is always being expressed relative to the unit vector, and also in a specified direction. Our gradient vector always tells us what specific direction to move in to find the highest point, because its magnitude is always equal to the rate of change, with respect to distance, in that given direction and along that specified axis.
We can now define a circuit in terms of these gradient vectors. Each circuit will have its marker points, which are its minima and maxima for each of the x, y, and z. These will always exist. That is to say, no matter what specific route we pick, every circuit will have its most easterly and most westerly points; and its most northerly and most southerly; and its highest and its lowest points. These minima and maxima always exist per each circuit, even though their values may change from one to the next. Thus although, for example, the mass flux implied by the divergence in mass of Maxim 1, Proviso 2, ∇ • M → 0, contributes to the biological potential by insisting that there be increases and decreases in the divergence or average individual mass, m̅, no explicit value now needs to be declared. It is sufficient there be an oscillation in mass. This is now a part of the biological potential. An infinite variety of circuits is now available. Each of them pursues a different trajectory. Yet all of them satisfy the same overall biological potential, which itself ranges, as a gradient, from a minimum to a maximum value and returns without ever having to demonstrate the same value.
We of course now have a contradiction we must resolve. It is an issue of the relative and the absolute. We have, on the one hand, the main declared purpose of all units for the standards of measure used within science. This is the most clearly stated by the Bureau International des Poids et Mesures:
The task of the BIPM is to ensure worldwide unification of measurements; its function is thus to:
 establish fundamental standards and scales for the measurement of the principal physical quantities and maintain the international prototypes;
 carry out comparisons of national and international standards;
 ensure the coordination of corresponding measurement techniques;
 carry out and coordinate measurements of the fundamental physical constants relevant to these activities (BIPM, 2006).
But we then also have, on the other hand, the sharp contrast provided by the oftcited “complexities” within biology and ecology which many feel belie any and all attempts to bring order and regularity to these sciences:
Strong observed that ecology differs characteristically from physics in that physics has rather few types of fundamental units—particles—and that these differ in relatively few ways (mass, charge, velocity, etc.). By contrast, the fundamental units of ecology—individual organisms—differ in many ways. Communities, composed of populations of different species, each itself comprising many different individuals, are vastly more variable still. It is no surprise that the complete, detailed dynamics or even the statics of communities have not been subsumed under tractable general laws, other than laws that govern their constituent elements (in the limit, the fundamental particles of physics). Lawton (1999) admits the possibility that we simply have not yet had sufficient imagination to have deduced more useful and interesting laws, but he is skeptical about our future prospects, and I agree with him (Simberloff, 2004).
The BIPM’s SI units are the backbone of science. Yet somewhat surprisingly, the BIPM itself gives hope to our cause by expressing its remit with a biologicalecological metaphor that even references Darwinian evolution: “the SI is, of course, a living system which evolves, and which reflects current best measurement practice” (BIPM, 2006).
It is initially difficult to know how a measure can be both specific yet malleable for almost by definition, a standard of measure must be constant. The central issue for biology and ecology is very simply and clearly presented by Dinerstein in his Effects of Rhinoceros Unicornis On Riverine Forest Structure In Lowland Nepal, a study of the rhinoceros’ grazing habits. Habitat destruction has placed the species under pressure, but in spite of those severe stresses Dinerstein’s study sheds light on the fundamental evolutionary issue confronting this species, and which our proposed unit normal vectors must squarely address:
The purpose of this study was to elucidate how large mammalian herbivores influence forest structure and canopy composition by inhibiting vertical growth of saplings that are frequently browsed and trampled. Specifically, I ask whether chronic browsing and bending of Litsea monopetala (Roxb.) (Lauraceae) by greater onehorned rhinoceros prevent most Litsea individuals from reaching the canopy.
…
Browsing and trampling stimulated production of new leaves and stems below 2 m. Saplings chewed and pruned by Rhinoceros produced significantly more leafy branches below the browse line than did unbrowsed effects saplings …. This is because most of the new growth on protected saplings was distributed at the upper edge of the crown, thus above 2 m, instead of at the base of the tree. Increase in leaf abundance below 2 m on browsed saplings is related not to phenological changes induced by herbivory but rather to manipulation of branch height and growth.
…
Even at reduced population levels, the interactions described here between Rhinoceros and woody plants clearly suggest a significant evolutionary impact of selective browsing by large mammals with potential cumulative effects on forest structure and canopy composition (Dinerstein, 1992, pp. 701704).
This is a completely unremarkable form of ecological complexity. As Rhinoceros unicornis browses, it affects the future of its own resources, and therefore its own future evolutionary potential. That potential will be expressed in the effects of the energy thus made available, through that specific browsing, both to itself and to its progeny. This reciprocity must be accommodated. Two things are therefore necessary: firstly, we have to determine the unit normal vectors that can act as standards of measure for biological populations; and then secondly, we have to show how they can be applied so they are malleable enough to be useful over evolutionary time.
As we did in our Brassica rapa experiment, it is invariably necessary to take several generations to determine the values applicable to populations. The greater the number of generations, the more accurate the value. For example, the generation times we measured for Brassica rapa were 44, 35, and 28 days respectively. B. rapa might be semelparous, but these times clearly shortened as the population density increased. There were also, of course, associated changes in mass, number and energy.
Whatever may be the realities of evolutionary diversification, the only possible cause for such differences is (a) the number and types of components, (b) the relative number densities of entities over those components, (c) their configurations, and (d) the resulting partitioning between work and heat. All these are properties of DNA.
We have in fact already met our first unit vector normal. It is the one biomole of N = 1,000 entities we have defined using both the Avogadro and the Franklin constants. We also carefully created an Euler function whose values we can scale arbitrarily, and without fear. Our unit value therefore always relates to a specific number of biological entities. As a point of principle, no matter how large or small any population may be, as soon as it completes a generation, it can return an average number over that generation. That average can then be scaled such that the number densities, over the cycle or time period, then oscillate around a 1,000 average … which we have defined as one biomole. And since we have already settled that our average over the generation length is one biomole, then that one biomole is n̂. It is the size of our unit vector with respect to our line segment of molecules. The unit vector can be used for integration and differentiation because it has a sense of direction around unity.
Therefore: there in principle exists an “engenetic equilibrium” which is a contemporaneous population of individuals selected from all points in the biological cycle of a given species such that progenitors and their progeny are jointly present and also such that their distribution by age is the equilibrium distribution for that species. Important biological variables can then be derived that are held in common by all within the population. As per Maxims 2 and 3 of ecology, the maxims of number and succession; and also as per Law 4 of biology, the law of reproduction; there is no entry into an engenetic population other than through those interactions amongst and between its members that directly produce progeny. An engenetic population is then and also the biological equivalent of the isolated system of thermodynamics in that it is the measuring stick against which all populations, and the variations in their numbers, can be measured. A “unit engenetic equilibrium population” is then, and by definition, an engenetic equilibrium such that the number of entities constituting the equilibrium distribution averages exactly one biomole over the course of the cycle.
The question of numbers settled, the issue then arises of establishing the time period for such a distribution. The issue is of course dominated by the fact that all organisms face the same ultimate difficulty. It is expressed in Proviso 1 Maxim 1, the maxim of dissipation: ∫dm < 0. All populations are at constant risk of extinction. But that risk seems to be related much more to generation length—i.e. the measure of temporal probability regarding a successful handoff of usable components—than it does to body size or to any other such allometric criteria (Martin and Palumbi, 1992).
The Verhulst equation of population dynamics, which in its highly simplified form is dN/dt = rN(1  (N/K)), led to the r/K selection theory (Begon and Mortimer, 1986, pp. 164–172). Evolution is driven, on this view, in one of two directions: either towards r, which is to rapidly increase population numbers, N, and to so fill the environment’s carrying capacity, K, as rapidly as possible, but while running the risks associated with wild oscillations in numbers; or else alternatively towards K, which is to invest more heavily in fewer offspring and so to remain close to that capacity for far more extended periods, which then runs grave risks when the environment changes—as is inevitable—and becomes adverse. Each strategy enumerated in the rK dichotomy of course has its costs and its benefits.
As is wellknown, rK selection theory does not give a full accounting. “The explanatory powers of the accepted scheme based on the r/K dichotomy are obviously limited” (Begon and Mortimer, 1986, pp. 169). The main point is that however it is accounted for, each population simply chooses a different competitive response to that same overall Maxim 1 proviso. Whatever may be the selected response, it is ultimately based on the mass, the energy, and the numbers of both microscopic and macroscopic components. Biological entities straddle that temporal divide between the microscopic and the macroscopic, and this should surely be recognized. That time span is the generation.
Our engenetic approach, with its line segment of molecules, differs from all others. It explicitly incorporates molecular behaviour. It also—and again explicitly—links those molecules to the generation. It thus ties the microscopic to the longterm evolutionary behaviour of organisms through the forces and potentials of the one upon the other, and as are built upon our line segment. On this approach, a generation is not marked by a specific time period. It is instead marked by a mass and an energy signature based on a circuit of biological potential. Its defining characteristics are work and heat. A generation is the absolute time span between repetitions of states as a population completes a circuit of biological potential and moves from one relevant and malleable minimum or maximum to another equally malleable one.
Both the microscopic and the macroscopic behaviours elicited over a generation can always of course be measured in absolute chronological terms. But such environmental cues can only evidence themselves through the work done and the heat emitted as one generation hands on its inheritable properties to the next. When measured with absolute time scales, any resulting diversification within phylogenies certainly provides vital information. But those absolute time scales again miss the overtly reproductive component: the periodicity and intensity of biological behaviour over that time span.
Many inheritable traits certainly take a relatively lengthy absolute period of time to work through, but they can also only happen relatively: i.e. through generation after generation. Relative time scales—both across and within species—are therefore of the essence in framing and analysing suitable evolutionary responses. The mass, the energy and the number densities used to traverse a generation—and expressed per the generation—must clearly then be primary objects of study.
Whatever the realities of evolutionary diversification, its only possible cause is (a) the number and types of components, (b) the relative number densities of entities with regard both to each other and to those components, (c) the configurations given to those components, and (d) the resulting partitioning between work and heat. All these will be properties of DNA. The generation length must express biological potential in terms of masses, energies, configurations, and rates of change thereof. This is the essence of evolution through natural selection.
Our directed line segment of molecules clearly holds the key to determining generation length. It ranges from t = 0 through to T … which is our intended generation length however we choose to define it. Since all biological behaviour is given by the general expression f(N, q, w), then the gradient ∇f = (∂f/∂N) n̂ + (∂f/∂q) q̂ + (∂f/∂w) ŵ defines the workings of biological potential, µ, over T. Changes in that gradient define a generation through an increase and a decrease in: (a) the numbers of entities maintained per second; (b) the numbers of components maintained per each entity; and (c) the amount of processing given to each molecule per second. The generation length—as a circuit—arises from the interplay of these relative increases and decreases.
We again take up a given number of biological entities from a unit engenetic equilibrium population. They will again immediately distribute themselves upon our line segment at a specific number density. We will again have trails of molecules of specified quantities, and all with the distributions appropriate to these entities.
If we observe a relevant number of biological entities for a stated period of time, we will see some entities being lost, and then others replacing them—within that time period—so that the average number of entities stays at one biomole, and there is no net increase or decrease. We will see a similar cycle for the moles of components of which they are composed, and also for the energies with which they are configured. Since all our time periods are now stated per biomole, and so by proportion of both time and entity, then all values for mass, for energy, and for rates will differ systematically across populations and their entities. And since given time periods will represent different proportions of respective life cycles, and also involve different numbers of entities and components, then this energy interaction suitably specifies the constraints of constant propagation, constant size, and constant equivalence.
The unit vectors or weighted averages for Brassica rapa, along with the minimum and maximum values, were determined from our experiment as:
Numbers 
Average individual mass 
Visible presence 


Unit normal 
1.000 biomoles 
9.248 x 102 grams/sec 
8.970 x 103 grams/joule 
Minimum 
0.662 biomoles 
1.171 x 103 grams/sec 
6.740 x 103 grams/joule 
Maximum 
1.096 biomoles 
1.049 x 101 grams/sec 
1.120 x 102 grams/joule 
We can now use the unit normal vector to define a potential for the number density all over the generation. This is the “engenetic burden of fertility”, φ. It is calculated by taking the inverse ratio between the biomoles of entities, N, at any point and n̂ : i.e. φ = n̂ /N. It is a relative measure linking the number of entities in biomoles at any point to the average number, N’ = 1 biomole. It therefore automatically includes a relationship between progenitors and their progeny at every t over T. The greater is the difference between Nt and n̂, then the greater is the magnitude of the force—the steepness or intensity of the gradient—striving to carry Nt back to n̂ at that t.
This engenetic burden of fertility, φ, gives a direction and a magnitude to N at every point. It expresses the direction of the maximum gradient, and therefore the operative force on both the molecules and the entities in moles and biomoles, and in terms of number. It is the measure of the tendency for the molecules composing any given entities to restore the population, at that given point on that line segment, and to carry it back to unity. Thus the entities strive either towards or away from n̂ and entirely in response to N’s value.
As per the fourth law of biology, the law of reproduction, the engenetic burden of fertility is an intrinsic property. If a species or population is to continue then all the entities together share the ‘obligation’ of ensuring that each is replaced by another like itself. Law 4 states that there must be at least one path in the allowed set that allows some entity to stretch out to T, and so therefore at least some amongst the members must meet this obligation. So if the numbers decrease, then the obligation imposed upon the others to succeed in reaching T increases … and they must therefore increase their energies commensurately over every remaining t.
It is again not required of a population that any randomly selected entity makes efforts to replace itself. All that is required is that the population at large ultimately replace all of its members such that the engenetic burden of fertility, when averaged out over all entities, is unity … which is the case for an equilibrium age distribution population. The entities that meet the said obligation to reproduce must therefore do so on behalf of all, or the species cannot continue. This is the intent of the engenetic burden of fertility, and it allows us to identify the locus of reproductive energies upon the line segment of molecules.
An engenetic burden of unity, such that φ = 1 means that there is a 1:1 ratio between all existing and all future entities over the population. By analogy with the hill, it means that the numer of hills and valleys traversed are equal, and we return to the same place on each circuit. If one biomole of entities happens to exist at any point, then each entity is on average being asked to do the work of exactly one biomole of entities per each unit of time; no more and no less. If this occurred all over the generation then the net gradient would again be zero everywhere. It thus represents the unit normal. It is the ‘normal’ rate of maintaining Q = 1 biomoles per second over all possible species and populations, and irrespective of their masses and compositions. It is therefore a general measure having application to all populations. This is the meaning of both the unit normal vector and the engenetic burden of fertility, whose dimensions are seconds per biomole.
Let now half the entities be lost so we now have N/2 biomoles. The engenetic burden of fertility will increase to φ = 2 seconds per biomole. The meaning is that the N/2 survivors must now corporately shift from doing only one second’s worth of work per each second, and per each existing entity, to doing two seconds worth of work per each second, and per each survivor entity. The survivors must in other words compose themselves to do one second’s worth of work each for themselves; and an additional second’s worth of work to cover the 0.5 biomoles that have been lost. The unit normal vector thus governs the rates of change and magnitudes of numbers of entities, and therefore of their molecules upon the number line. And since it represents the long term equilibrium age distribution, then it is an influence upon each entire generation length, T. It states the potential’s precise magnitude and direction.
We can now apply these unit normals and the above principle of the engenetic burden of fertility—which is simply the directional derivative extracted from the gradient on our line segment of molecules—to our Brassica rapa experiment.
The various values for the Brassica rapa numbers give it engenetic burdens of fertility of φ = 0.912 and φ = 1.510 seconds per biomole respectively at its minimum and maximum values. The minimum value is at the seed stage and is below unity. It is a directional derivative informing us that the actual number density is greater than the average. Therefore, the engenetic burden—the potential force active upon this portion of the line segment relevant to numbers—is low, with each entity being under a relatively small obligation to work at maintaining the population for their numbers are in excess. Each is therefore free to reduce its work rate to a point below the unit normal vector whether this be in mass, in energy, or in both.
The fruiting stage has φ = 1.510 seconds per biomole. This implies that the numbers of survivors is now less than the unit normal, and therefore below optimal. Each surviving entity must now work proportionately harder than the normal rate to compensate for the numbers lost. Each of the survivors must therefore process molecular components at oneandahalf times faster a rate per each second than is expected for the unit normal of the flat gradient. The fruit is therefore operating at a rate 2.28 times greater than the seed. This is in part because of the increased mass, and in part because of those lost entities. Our gradient thus has both magnitude and direction at all points in all circuits of biological potential.
The same principles that we have used for n̂ can now be applied to m̂ as a unit normal vectors for mass, and as t moves to T. There is always a divergence or average individual mass, m̅, over the population at every t. But by Maxim 1 Proviso 2 of ∇ • M → 0, there is also an m̂ operative at every t pulling all m̅’s towards it and which is a property of the entire generation, T. This is a distinguishing feature of the cycle. The magnitude of m̂ at work on m̅ again gives m̅ a relative magnitude and sense of direction, and states how strongly the entities creating that m̅ react to that unit normal vector of m̂ that is a property of the generation constantly at work upon them all. And as generation follows generation the value for m̂ will ever more closely describe the average value for each generation, and so also for each entity—as also the forces at work for each m held at each t, and when related to that m̂, the unit normal vector for m.
As in Table 4, we can find those normal values and turn them into units by scaling through multiplication. Thus although we continue to use the term unit normal vector, it should be understand that it is rather a reference point for a standard of measure relative to a given species or population. Its value does not have to be unity at that reference point. It can produce unity when divided at that point by the value we assign to the unit normal vector. The values below and above unity then have a sense of direction, as well as a magnitude.
The expression ∫M dT describes the population’s mass over the entire generation. It will also return an M’ as an average rate of propagation for the population. However, the expression ∫dm̅ dn also describes a sum over the entities’ mass. It will in its turn return m’ as an average individual mass over that specified generation. And as generation follows generation the value for m’ within each specific generation will ever more closely describe the average value over all generations, which is m̂. Our Brassica rapa experiment again brought several generations worth of m’ together to determine the unit normal vector, m̂, given in Table 4. And since this m̂ or unit normal vector is a property of the species, being taken over many generations, it establishes both a scale and a direction for the force at work upon mass at each t and for each n over any and all Ts.
The “engenetic burden of components mass”, κ, is now calculated by taking the direct ratio between the mass of the entities at any point and m̂ : κ = m/m̂. It is again a relative measure linking the mass of the entities to their average mass over the many generations they have been measured, which is the value for the equilibrium age distribution population. At each and every t over T, the n entities on our line segment of molecules will have an average individual mass over the population, of m̅t. This estalishes their q̅t moles of chemical components. All these values “belong” to that specific t, in that specific generation, and as the entities interact with the environment through their escaping and their capturing tendencies. Generation after generation, every entity at that t will be affected by the work needing to be done at that t, but also as affected both by the environment and the others around it. But since the entities are distributed by, and according to, n̂ —the unit normal vector over numbers—then there must be an m̂ which is in its turn operative on the entities at each and every t, and as they gather specified components. That m̂ expresses how any given m̅t relates to that m̂ for that entire generation, and also relative to other generations and so the equilibrium age distribution population. Thus Brassica rapa’s initial average individual mass is 1.171 x 103 grams per second while its final average individual mass is 1.049 x 101 grams per second. When these are applied to the unit normal vector of 9.248 x 102 grams per second, we produce κ = 0.0127 and κ = 1.134 respectively. B. rapa’s initial mass is quite considerably below that of the unit normal vector—about 1/100th of the value—and it is therefore required to put on mass at a fairly rapid rate per surviving entity.
As with the engenetic burden of fertility, the magnitude of the difference between m̂ and m̅ states a relative magnitude—and thus a scale and a sense of direction—for the mass of components absorbed or released. It states how strongly the entities creating that average individual mass are relating to that unit normal vector, and so how much more quickly or slowly they will put on or take off mass over the given generation, but always as compared to that unit normal vector for mass.
All these same things again hold for v̂ and the visible presence operative at any given time. There is a v’ as the average individual visible presence for any given generation, with v̂ being the unit normal vector for all those v’s over many generations, and as describes an entire species through its equilibrium age distribution population. The “engenetic burden of conformation”, χ, is given by the inverse ratio between a given V and v̂ as in χ = V/v̂. The minimum and maximum values for Brassica rapa’s visible presence are 6.740 x 103 and 1.120 x 102 grams per joule respectively, giving χ = 0.751 and χ = 1.249 again respectively. Thus the zero value for the divergence in mass and energy is again both constant and malleable, but is also always measurable. This is exactly the result we wanted to achieve.
These three engenetic burdens of fertility, of components mass, and of conformation—φ, κ and χ—each relative to their unit normal vectors, now establish the rates of operation upon all entities on the number line of entities and their molecules to produce the characteristic behaviours and features of each species and population. This is the biological potential. And since µ is some function of φ, κ, and χ—whether additively or multiplacitvely—then biological potential, which is the rate of change of engeny and is measured in watts, can now trace endless paths over and across this landscape of biological energy … producing those variations in mass, in energy and in number as are at one and the same time both characteristic of, and causes of, natural selection, competition and evolution. We have now achieved our purpose of describing the normal behaviour of natural selection in a scientific and rigorous fashion. The span of a generation—and therefore the field of natural selection—is a relative value of one rotation of biological potential through all those different values. Our last step is to give this an absolute value, and the exercise will be complete.