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# 38. The third maxim of ecology: the maxim of succession

The total of the resources and chemical components that a population removes from and returns to the environment, using its mechanical chemical energy and on account of its escaping and capturing tendencies, is its mass flux, M. There is a mass flux at each and every t over T. It is also given by M = n x m̅ and so depends on the number of entities, n, being maintained at any t. Those n then establish the divergence of that specified mass flux, which is the average individual mass, M/n or m̅, for that t. As per the second maxim of ecology, which is the maxim of number, we can always count a specified number of biological entities as are responsible for any specific mass flux.

But by Maxim 1, Proviso 2, of ∇ • M → 0, the average individual mass, and therefore also the total mass flux, must constantly oscillate around the respective and specified generational averages, m̅’ and M’ respectively. Thus the divergence in the mass flux can be either positive or negative depending upon whether m̅ is increasing or decreasing. Since the mass flux is M = nm̅, it can change when: (a) the population’s average individual mass or divergence changes; and/or (b) the volume, which is the population count, changes.

- Maxim 3: Proviso 1

Since the divergence is given by m̅ = M/n, then the first way a divergence can increase is for the mass flux to increase. The visible presence is the energy density and is given by V = M/P, or else by m̅/p̅. If the population’s divergence or average individual mass, m̅, increases while its visible presence remains the same, then the average individual energy, p̅, must increase commensurately. Since the stock of chemical components increases by Δm̅, then so also does the mechanical chemical energy needed to bind them increase by Δp̅ which is by proportion. Therefore, the net energy attributable to all the entities also increases. And since these components and energy increases impinge upon point-entities, then the net circulation of energy about them increases and the net processing increases. The curl or vector cross product, ∇ x, has therefore increased.

The curl or circulation or rotation about any point of interest is a form of rotational kinetic energy. It determines the total energy attributable to any point. An increase in the curl is an increase in the circulation of flux about those point-entities, which thus increases their abilities to cause transformations about themselves. Therefore, any time-rate of increase in the average individual mass of chemical components held per second by a population—which is ∂m̅/∂t—causes an immediate increase in the population’s energy. The intensity of their field lines increases, and the net energy flux passing through our Gaussian surface also increases. And since this is an equilibrium age distribution population with stationary states that also lie upon paths, then any such instantaneous increase in mass is a definite change in each entity’s state at any time, t. Each entity therefore arrives at the following t with both that increased mass, and that increased energy intensity as is associated with its increased curl or circulation in energy about itself. Therefore, an instantaneous increase in average individual mass, ∂m̅/∂t, causes the entities to increase their energies by an amount that depends entirely upon the time rate of increase, ∂p̅/∂t. The curl or vector cross-product thus augments the increase in divergence by increasing each entity’s throughput of energy and work. And since this is an equilibrium age distribution, then for every increase in the curl at any given point t, there will be a matching decrease at some other point or points along the generation length, T, to maintain the stated equilibrium. These accompanying and compensating increases and decreases in the circulations about boundaries at every t thus sum together to create the relevant equilibrium distribution of mass and work over all ts over the entire generation length, T, and so also from one T to a successor T. We in other words have a transmissible change induced from one generation to the next and that is also a measurable force as a circulating mass flux for generation after generation … and in the precise manner stated by Darwin:

- +∂m̅/∂t

- Maxim 3: Proviso 2

Since the divergence is again given by m̅ = M/n, then the second way a divergence can increase is for n, the volume or the population count, to decrease. But each individual entity amongst the n contributes to the net size of the sphere that represents that population, and that emits the mass and the energy flux of concern. These population numbers therefore also determine the circulation of energy about the surface. And if the numbers decrease, then the net volume decreases … and we have already determined that this immediately increases the divergence or the flux per unit volume. The sphere’s surface area also decreases which immediately increases the curl or the circulation about that sphere per unit of its area. And since this circulation is the force that pushes the flux about the entire boundary, with the curl being the amount of swirl or circulation per unit area of its surface, then the curl increases as soon as the numbers decrease. Furthermore, the curl is increasing in direct opposition to the decrease in numbers. It in other words strives to restore the system.

Since we have here, in a curl, a force that tries to restore the system as n has decreased, then at any and all ts over T, and whenever the numbers in the population are prone to decrease at the rate of -∂n/∂t, there is an immediate force instituted over all the remaining n to move oppositely and to increase and by exactly that same magnitude. As Charles Darwin therefore declared, an entirely transmissible change in energy and behaviour is induced in all extant survivors; which is then transmitted to all successors; and entirely through the mass of chemical components they will all hold per second at that following t. Thus a change in population number, n, is independently capable of inducing an immediate—and opposing—change in biological systems and their entities, and we therefore have:

- -∂n/∂t.

Since this mass flux of M kilogrammes per second circulates at every t over T, then its circulation about all the n entities over that T is given by ∫M dT. This circulation of mass is then the force whose energy both acts on the population … and that the population in its turn acts upon in order to survive and to reproduce itself. It circulates about the population’s boundary at a given rate in kilogrammes per second over that entire generation length, T. This integral of ∫M dT thus produces the total mass used by that population over that time interval or generation length, T.

At each and every t over the whole of T, each point or biological entity hands its mass on to its successor at the next time-point, so that that successor can then and in its turn radiate the necessary and expected energy flux for that t. The amount of energy radiated in that flux will depend upon the quantity of mass, m̅; how it is configured; and how many entities are involved in producing it.

Every distinct entity within the population contributes to the circulation evident over T. Each entity’s mass is m. There are n in the population. If we take a counter, i, and range it from i = 1 to i = n, then the population’s net circulation is given by the mass, mi and the individual processing rate or visible presence, vi, of each of the n entities. While each of m and n contribute to the mass flux, M, they do so independently and in direct proportion. Given that every individual entity contributes to that mass flux, then each entity’s contribution, which is the circumference or surround or area of record per each entity, can be determined from ∫dm dn over the population. And since a first commodity, m, is being multiplied by a second, n, then this is two-dimensional and so is effectively an area … which is what we need for our planimeter.

We can now take each of our distinct entities from i = 1 to n at any given t over T. We can sum the masses of all the individual components that each entity processes over those various ts, and we can then again sum all those sums over all the n entities involved in processing the population’s entire mass circulation, M, all the way around T—which is the meaning of ∫dm dn. And since these individual masses can, without error, be regarded as evenly distributed over the n entities, then the integral can equally well be reckoned as ∫dm̅ dn. But … this last integral of course produces the total circulation over the generation … which we have already determined to be ∫M dT. And since ∫M dT = ∫dm̅ dn, then this is little more than M = m̅ x n.

Since we now have expressions for both the circulation and the area over which that circulation passes, we can produce an expression for the curl, which is the circulation per unit area. The curl simply tells us how rapidly the entities that the given circulation is passing over are each independently processing that flux as their contribution to the net circulation about their common boundary. The curl thus says something about the way in which the entities are configured, and about the ongoing chemical reactions. And since the curl is simply the cirulation per unit area, then it is immediately given by ∫M dT/∫dm̅ dn. This curl in other words gives information about the nonmechanical chemical energy and its necessary behaviour as m and n change, the one directly and the other inversely.

Our expression for the curl implies that the larger is the number of entities processing any given circulation, then the more slowly they each do so per unit time, and so the smaller is their nonmechamical chemical energy … and conversely. But since we have M/m̅ as a determinant of the curl, then the rate of processing or nonmechanical energy is again dependent upon number, for M/m̅ = n which is simply the volume over which the flux passes. Therefore, the numbers in the population at any t over T cannot be separated from their metabolisms and their physiologies, nor from their inheritabilities … which is exactly what Darwin declared.

We can now combine our two changes in population mass—one in m̅ and one in n—to give the total curl which is the vector cross-product for the population’s mass of chemical components held per second, M, at any and all ts over the entire generation, T. The third maxim of ecology is the maxim of succession and declares the Darwinian principle of the succession or transmissibility of biological induction … which is precisely evolution … and which is:

The Third Maxim of Ecology: The Maxim of Succession

∇ x M = ∂m̅/∂t - ∂n/∂t

[Darwin’s theory of competition]

The rate at which progeny is produced depends upon the rate at which competition occurs.