46. Steps two and three in defining a species: the field of natural selection

Flexibility in our systems of measuring is all very well and certainly helps us in accommodating the extent of the variations that populations and species display. However, we have yet to determine what they share: what is being repeated, and exactly what attributes lead to the periodicity we are striving to measure. Additionally: the Liouville theorem tells us that in spite of all their variations a given population of entities holds a phase volume in common as it maintains a given energy boundary and/or maintains a surface that can surround a volume. We can even conclude the existence of space and time averages for all the neighbourhoods concerned, as populations establish common values for their total fluxes of both mass and energy within those volumes, and about those boundaries. However, this does not specify the properties the populations hold in common within those boundaries and volumes.

Suppose, upon our earlier Africa-Asia survey, that we covered one thousand miles and observed a set of biological organisms that look similar to, but appear to have no direct genetic contact with, a population we saw earlier. How are we to determine whether they are, or are not, the same population or species, despite that apparent lack of genetic contact, and even though they might seem to share a common generation length, habitat, and other such values as contribute to totals of their masses and their energy fluxes? The Liouville theorem could seem to hold between them for they could easily have similar generation lengths, and seem to share given traits in common … but so do many populations that are not in fact species. The Liouville theorem does not tell us what to measure to guarantee equality. It only holds for entire neighbourhoods of points. It says nothing about specific points or populations and their trajectories. This is not just a problem of populations over space. It is a problem of generations over time.

Figure 54: The Equality of Natural Selection

We are using vectors to measure all our biological properties. We have already seen that we can specify a generation length or period through our vectors simply by specifying a ratio between any two chosen properties. We can thus refer to a first population via the vector P1(n, q); and to a second as P2(n, q). Since, by the Liouville theorem, they will share the same phase volumes and boundaries, then these together determine the net mass and energy fluxes at each t over T. As in Figure 54, that establishes the net flux passing through our Gaussian surface. If the two populations—P1 and P2—that we have under examination are the same, then their respective vector fields must be identical in every relevant particular, and at every point. They must, more specifically, evidence the same number densities and and the same moles of chemical elements at every t over T … and so therefore the same fluxes as those ratios—i.e. angles—oscillate and repeat. This is the overall pattern we successfully created for the Liouville theorem.

If we should choose to switch our variables, we still want the vectors we are using to describe the two populations to be identical. Therefore, the issue of identity in vectors is a more general one. If vectors can indeed be shown to be identical no matter how the objects they describe are transformed, then we can rest assured that each and every change in property is due to some real change in either the entity under observation, or else in its environment, and is not being caused by some theoretical or mathematical deficiency. We can then have greater clarity and make some firm pronouncements about natural selection irrespective of specific properties and ratios.

Whatever the size of the flux emitted by P1 at any given time point t, we want P2 to emit the same flux at that same time point and so that P1(n, q) = P2(n, q) for every t over T. We can now turn to Gauss’ theorem, the divergence theorem, and one of the four fundamental theorems of the vector calculus. It links areas to volumes. We do not yet know what forces, F, are active around the boundaries of these two populations, nor the precise properties of the flux passing through their surfaces, but the divergence theorem tells us that they must both satisfy the condition that:

∬∂R F • n ds = ∭R ∇ • F dV.

But not only must the two populations satisfy the divergence theorem, they must also satisfy Green’s theorem which links lines to areas. This is that both:

∫C F1 dx + F2 dy = ∬R ((∂F2/∂x) - (∂F1/∂y))  dA,

and

∫C x dy = -∫C y dx = ∬R   dA.

These three conditions give us another way of expressing our requirement that our two population vectors P1 and P2 be equal (Dash and Khuntia, 2010, p. 48; Carlen, 2007; Fitzpatrick, 2006a; Potter). Since the forces around their boundaries must be equal then their divergences and their curls over their entire generations must also be equal at every point. In other words, their fluxes and their circulations—which are the forces they experience due to their volumes, surfaces and boundaries—must be the same. If curls and circulations or fluxes and divergences are ever different between these two populations, then those two populations cannot be the same. Therefore, their divergences and their curls must always be equal.

We first consider the divergences. If P1 and P2 are equal, then the vectors representing them must have the same relationships to their respective unit normals drawn upon each of their surfaces. That is to say, the differences in their divergences must be zero. We must in other words have ∇ • P1 = ∇ • P2 . But this is simply ∇ • (P1 - P2) = 0. And since the difference between two vectors is just another vector, then this is exactly the same as asking whether or not it is possible for there to exist a vector field that has zero divergence.

We now look at the curls. If P1 and P2 are equal, then since their relations to their respective unit normals must again be the same, their circulations must also be equal meaning ∇ x P1 = ∇ x P2 . But this is immediately ∇ x (P1 - P2) = 0. So not only must there now be a vector field with zero divergence, it must also have zero curl so it contributes nothing to any circulation.

We are now in the position that two biological populations can only be the same if a putative or random or undesired vector field, U, can exist that is both “incompressible” or “solenoidal”; and “irrotational” or “without curl”. That is the only way that our new undesired vector field that is the expression of the difference between P1 and P2 can contribute nothing to either P1 or P2 at any point. This is the only way that the biological equivalences of our law 2 of biology, the law of equivalence, can be established. If biological entities and their ecologies are going to be unpredictable and disorderly, then let it not be because of the all-too-human predilection for etymological confusions.

Figure 55: Difference in Natural Selection

We now know that if this undesired vector field, U, can exist, then its divergence is the divergence of whatever undesired field we get by subtracting P2 from P1: i.e. ∇ • (P1 - P2) = ∇ • U. Figure 55 now shows the process of subtracting two vectors. We simply reverse the direction of the vector to be subtracted, in this case Vector 2; we then lay it at the head of Vector 1; and we shift Vector 1 by the given amount in that direction. The process produces our random vector, U, as their difference. Since it must contribute (a) no divergence and (b) no curl, then these must sum together to produce the entire field. We now need to demonstrate, for this field, that:

U = -∇ • U + ∇ x U

which says that this field must be the sum of its divergence and its curl, and that it must have no other properties anywhere throughout it. This must be its complete specification.

Now that we have our new undesired vector U; and now we know it must have no divergence and no curl; we can turn to Maxwell who has already taught us how to handle this within the vector calculus:

Now if σ be a vector function of ρ and F a scalar function of ρ

∇F is the slope of F

V∇ • ∇F is the twirl of the slope which is necessarily zero (Maxwell, 1870, pp. 568–569).

We now know, from Maxwell, that our undesired vector U must be the curl of the gradient of some random scalar field, φ, which must always be zero. This can be understood through gravitation, which has no curl. Planets in orbit do happen to always be rotating about their axes, but that is always due to some external factor. It is in principle possible to imagine an orbiting planet with zero rotation, such that it always keeps the same face to its sun in its orbit. Gravitation itself—which establishes a gradient for weight and for objects falling that varies with their height or distance—in itself causes no rotations simply due to that attraction. It only pulls gravitationally. It does not of itself cause spinnings or rotations. So our undesired vector U must now act similarly to gravity. It must itself be the gradient of some other function with zero curl. We must therefore set our new undesired vector field U equal to the gradient of some arbitrary function so that U = ∇φ. We must then follow Maxwell’s instructions and take the divergence of this gradient so we can establish the condition that it is zero. This then becomes ∇ • ∇φ = 0. And since ∇φ is both a gradient and a scalar, then its divergence is ∇2φ, which now means that we must have ∇2φ = 0 as a condition for equality between our two populations, P1 and P2.

And … this latest condition is a stroke of great good fortune because, as described by Carl Boyer, ∇2 is the “Laplace operator” and extremely well known in physics:

Typical was his study of the conditions for the equilibrium of a rotating fluid mass, a subject he had considered in connection with the nebular hypothesis of the origin of the solar system. … According to the theory of Laplace the solar system evolved from an incandescent gas rotating about an axis. As it cooled, the gas contracted, causing ever more rapid rotation, according to the conservation of angular momentum, until successive rings broke off from the outer edge to condense and form planets. … It is in this classic that we find, in connection with the attraction of a spheroid on a particle, the Laplacian use of the idea of potential and the Laplace equation. … Laplace developed the very useful concept of potential—a function whose directional derivative at every point is equal to the component of the field intensity in the given direction. Also of fundamental importance in astronomy and physics is the so-called Laplacian of a function u = f(x, y, z). This is simply the sum of the second-order partial derivatives of ∇2u, namely, uxx + uyy + uzz, often abbreviated ∇2u (read “del-squared of u”), where ∇2 is called Laplace’s operator (Boyer, 1991).

If the Laplace operator is good enough to handle solar systems, galaxies and gravitation, then it is certainly good enough to handle this trifling situation. It defines a harmonic function in two variables, and it assures us—in vector terms—that there are no arbitrary sources of either material or energy emanating from anywhere within the volume we are studying. There are no unwarranted sources or sinks anywhere throughout it. The Laplace operator states the net rate at which a substance moves towards or away from some point. Since it is declaring the flux density of our gradient, which is the first of our conditions, we can be assured that this unwanted vector field U, which is the difference between our two populations P1 and P2, has no divergence anywhere. The Laplace equation confirms that any contribution our undesired vector U might make to the said net flux arising from combining our two populations is truly zero; again confirming the existence of the steady state for U we seek. Thus the average individual masses are the same throughout the two populations at every t over T.

We now have to attend to the curl, which is the energy that moves about our point biological entities, and that then creates the circulations of both mass and energy over the generation for each population. By a slight restatement of Green’s theorem we can relate the flux flowing through a given volume, V, to that entering and leaving its surfaces S, as follows (Dash and Khuntia, 2010, pp 48–50):

∮V (ψ∇2φ + ∇ψ • ∇φ) dV = ∮S(ψ∇φ) • dA.

where ψ and φ are continuous scalar functions and whose partial derivatives are of the first and second orders. Since we want the two functions or populations to be the same, we can now set ψ = φ to give:

∮V (φ∇2φ + |∇φ|2 )dV = ∮S(φ∇φ) • dA.

We can deal first with the right-hand side. We know that our undesired vector field, U, is the gradient, ∇φ, we now see in the integral on the right. If we make that substitution, we then have:

∮V (φ∇2φ + |∇φ|2 )dV = ∮S(φU) • dA.

But the right hand side now simply states U’s complete divergence … which we have already settled and determined, through the Laplace operator, to be zero, meaning that the whole expression becomes:

∮V (φ∇2φ + |∇φ|2 )dV = 0.

This now leaves the integral upon the left. We straight away and again see the Laplace operator. We know that the Laplace equation is zero: ∇2φ = 0. That therefore removes the left-hand term from the left-hand integral to give:

∮V  |∇φ|2 dV = 0.

And this takes us virtually to our target. That expression in the integral is simply the square of ∇φ, which is again the gradient of our undesired vector field U. This at last gives:

∮V  |U|2 dV = 0.

And that zero is exactly what we are looking for. The above expression means that our undesired vector field U is contributing nothing to the curl, which are the fluxes that flow around our biological entities.

We have now settled that our undesired vector is truly a zero vector absolutely everywhere throughout the entire volume V bounded by our Gaussian surface, which encompasses an entire generation. It adds no divergence and no curl anywhere. And if this field that is the difference between P1 and P2 is zero absolutely everywhere, then P1 and P2 are the same everywhere. And since we proved this identity by determining the value of both a divergence and a curl, then any vector field can be rigidly and tightly defined simply by stating its divergence and its curl.

Two conditions must in fact hold before what we have just proven can be true. But they are the same conditions that allow the Liouville theorem to hold true so it can in its turn specify the boundary conditions applicable to these populations or vector fields, and so that they can in their turn specify the individual properties that must hold within the phase volumes and overall energy conditions that the Liouville theorem in its turn establishes for them … so we have already defined our population vectors to meet these conditions.

The boundary conditions for these two theorems, and for these population vector fields, state that as vector fields tend ever closer to infinity, then their curls and divergences must approach zero more rapidly than 1/r2. In other words, by the time we have reached r = ∞, we must also have [∇ • U]∞ = 0 for the divergence and [∇ x U]∞ = 0 for the curl (Weisstein, 2011c). And … these two boundary conditions together with the U = -∇ • U + ∇ x U we have just proven are in their turn the Helmholtz theorem which is the fundamental theorem of the vector calculus. It guarantees the uniqueness of all vector fields granted these boundary conditions. The Helmholtz theorem states that any vector field can always be decomposed into its sum and its curl, and that the decomposition is unique, given those boundary conditions. No two vectors can have the same divergence and curl without immediately being identical.

We have deliberately constructed our model to abide by the Biot-Savart law. Thus the stipulated boundary conditions needed for both the Liouville and the Helmholtz theorems are met by biological populations. They are built entirely on flowing electrons. These are by definition electric currents. The condition 1/r2 is the divergence of an electric field. It is also the condition for the divergence of the induced currents of the point-elements-of-current-segments. These not only have divergence, but also direction and distance. These boundary conditions simply mean that as we remove biological entities from their sources of electrons, then their energies must similarly approach zero, and they must dissipate and release their molecules back to the environment. If they were to dissipate less rapidly than their constitutive electrons, they would be in breach of both the first and the second laws of thermodynamics. Since the required conditions are met by our biological vector fields as defined, then the Helmholtz theorem is immediately applicable to biological entities. Every population that inherits a given set of values for its curls and its divergences in its mass and in its energy will behave identically to all other populations or generations also inheritng those same values.

By the Helmholtz theorem, biological populations must abide by the equations of continuity, meaning that the only means possible for populations to continue is for their members to use their masses and their energies to generate others like themselves. There is no other source of mass and or energy available to any biological entity or population … and this is how they are defined. We now only need to find this internal energy that is replaced which we have defined as biology, and the processes that facilitate it, which we have defined as ecology.

By our first law of biology, the law of existence, biological entities must always do work. As soon as they do not do work they cease to be biological … and that is certainly to dissipate more rapidly than their electrons. Maxim 1, Proviso 1 of ecology also states that ∫dm < 0, which is no more than a biological restatement of the second law of thermodynamics. Thus the boundary conditions for the Liouville and Helmholtz theorems are again satisfied. Biological entities must dissipate; they do not approach infinity; and there is some unique boundary condition for each specified one that establishes the energy volumes available to it once given its divergences and curls in mass and energy … and which reciprocally and also establish an energy domain.These are—simply—the average individual mass, m̅, and the average individual Wallace pressure, p̅, over the population.

We have now demonstrated that an entire population of biological entities—and so therefore a species—can be very tightly and uniquely defined simply by stating the average individual mass and the average individual Wallace pressure for that species over a generation, whose length we can also rigorously specify. No two populations can have these two values the same over a generation without being members of the same species … which is to have the same DNA identically configured and identically utilized over the entire length of that generation. This is a very considerable and very important result which we can now use to attack the problems of natural selection.