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# 36. The function for natural selection and evolution

The gradient theorem of line integrals is one of the four fundamental theorems of the vector calculus. It specifies behaviours along lines. The theorem thus relates a line integral directly to the function’s values at its boundaries. A gradient is such a line. Our two chosen endpoints can easily be the beginning and end of a generation on our line segment of molecules, or any t1 and t2 along T, the full generation length.

We earlier criticized Turchin for saying, in his Does Population Ecology Have General Laws? that “the similarity between the exponential law and the law of inertia is striking” (Turchin, 2001). It is one thing to criticize … another to propose a better model … and we now use the gradient theorem in conjunction with our line segment of molecules to begin building that better model.

We again refer to our equilibrium age distribution population. Whatever happens at any one time point t is related to what happens overall and at any other. We can determine the total volume or flux of all molecular components emerging from our line segment or molecular supply tunnel of molecules, as well as the biological entities in which they are incorporated, as in Figure 27, simply by counting. Once that count is expressed in Q biomoles per second, we can also and in principle use chemical analysis to determine the moles of each chemical component they each use per second. We can additionally measure the total energy contained both in their bonds at t, and that they expend in work and heat per entity and per chemical component. This will be a Wallace pressure of P watts over the population.

With that number of entities per second in hand, we now turn to Galileo and Newton. We take up our three constraints of constant propagation, constant size, and constant equivalence. As our biological entities arrive at the beginning of our line segment or tunnel of molecules, the behaviour we expect of them is the perpetual motion established by our three constraints. Those three constraints between them establish a given rate of work per second, a given joules per entity or per volume, and a given set of material components configured in a given way and as can compose that population of entities. And just as Galileo and Newton define mass and inertia as any deviation away from their definition of perpetual motion, so also do we define biological inertia as any deviation away from the values established by our three constraints.

Once they reach our line segment of molecules, our biological entities move steadily along it with the values for our three constraints. They now pass through every time point t to reach the terminus of the generation length T. As they pass over every t they must firstly use mechanical chemical energy to uplift necessary components into themselves; and they must then secondly use nonmechanical chemical energy to configure themselves so their rate of activity is as stated by the three constraints. They must do a net of work at those rates or the population and its entities will symbolically “fall” through the cracks and degrade.

Given these essentials then the behaviour of biological entities all along our line segment is defined by three variables:

- There is a given number of biological entities, n, at every t over T.
- Those entities are all composed of a given number of chemical components, q, in moles, again at every t over T.
- Those components are processed at a given rate w, in watts, per kilogramme, which produces a given amount of heat and work.

Since these are the only three relevant factors that affect behaviour on our line segment, there immediately exists a function in those three variables of the general form φ(n, q, w).

If the environment now imposes some change on these entities, then the environment will esbalish a potential. Our function φ(n, q, w) neatly allows us to determine the result. The simplest expression for the gradient, which is the potential, is -∇φ = (∂φ/∂n, ∂φ/∂q, ∂φ/∂w). This imposed change emanating from the environment means that the entities must now do extra work to revert to equilibrium … and the gradient tells us how much. By whatever degree one factor changes, the others must change by an amount to compensate for it or the generation cannot be completed. The gradient is thus a statement for how the population will respond as that potential is established … and therefore how they will respond to any change whatever. It is also measurable. We measured this precise potential in our Brassica rapa experiment.

The gradient is of course flat over the entire generation for a truly equilibrium age distribution population that abides continuously by the three constraints. That does not of course mean that the gradient is always going to be flat at all points in between for all populations. But all changes in entity numbers, ∂φ/∂n, in moles of components held per entity, ∂φ/∂q, and in net processing, ∂φ/∂w, will mutually affect each other, and will be mutually accessible from all three axes to restore the population should any change ever be imposed. Thus no matter how much the environment changes, the population can exploit the gradient to restore the equilibrium values by doing work upon the line segment.

If we want to know the overall divergence for the population, we need the net flux. The generation establishes the basic timetable, and we must therefore count the entities at both exit and entry points. Since an equilibrium age distribution population has an intrinsic rate of increase of zero, every entity lost is replaced. The numbers entering and leaving the generation are the same. Therefore the divergence or flux density holds the same value all over the generation. The number density at all points is equal, and the entities distribute themselves again evenly at every t over T. We have already established this figure as one biomole. And by the same reasoning, the numbers of components, and their rates of processing, will also be the same at every t. These are the three constraints of constant propagation, constant size, and constant equivalence.

And … we therefore now have our simplest possible and ideal population: biology’s equivalent of perpetual motion. This perfect biological population is one that averages one biomole at every t over T, which is its divergence, never deviating towards or away from that number, and so that N = 1 while d2N/dt2 = 0. Each of those entities is also composed of q moles of components at all times, again never deviating away from that number, and always such that d2q/dt2 = 0. And then finally, those chemical components are held together by a rate of chemical processing of w watts per kilogramme, again never deviating away from that number and such that d2w/dt2 = 0.

Just as Galileo and Newton’s is the perfect measure from which all real bodies deviate, this perfect biological motion is that towards which all biological populations in their turn tend whenever an intrinsic or extrinsic factor—or any other force—establishes a potential by taking them away from these steady state values. These are all the values around which they oscillate. We can now measure biological inertia, and determine the power, pace, and force of natural selection as the numbers in the population ebb and flow along with the components and their processing.