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41. An Euler equation for natural selection
µ 
= 
dS 
= 
(∂S/∂U)V,{Ni} dU 
+ 
(∂S/∂V)U,{Ni} dV 
+ 
Σi(∂S/∂ui)U,V,{Nj=/i} dui 
+ 
Σi(∂S/∂vi)U,V,{Nj=/i} dvi. 

1. Probundance, γ 
2. Procreativity, ψ 
3. Abundance, C 
4. Accreativity, Y 

Essential development, λ 
Compensatory development, L 
We now state a first population equation; explain its variables; indicate how they are derived; and then indicate the meanings and import of the various terms. We take full advantage of the properties of Eulerian homogeneous polynomials and functions of degree one. We also follow Gibbs and construct our Euler equation so that a set of partial derivatives describes and defines some suitably composed extensive biological variables that we attach, and whose partial derivatives immediately describe our variables and clearly delineate biological behaviour.
 The five variables
The five variables we need for our equation are µ, S, U, V, and N which we describe as follows:
 We call µ the “biological potential”. It measures Darwin’s ‘Natural Selection’ in watts. It is momentbymoment responsible for partioning energy across the population, and as detailed in our fourth maxim of ecology which is the maxim of apportionment.
 We call S the engeny or ‘inspeciesness’ of a population. It is taken from the Greek en or ‘in’, and genos or ‘species’. It is measured in joules per kilogramme per biomole per second and states the amount of energy that one biomole of a given population uses to maintain one kilogramme of its given numbers and types of chemical components for one second. It is alternatively measured as watts per kilogramme per biomole, or else as darwins per kilogramme per second. It is the combination we need of both intensive and extensive variables, while itself being an extensive one. It therefore states—and measures—all pertinent biological and ecological variables and can be used in all situations we must make a measurement … while also taking us straight to the behaviour of the molecules on our line segment or tunnel of molecules, for it also measures those molecules through their amount of substance in moles and then in kilogrammes. This is handled by our planimeter.
 We declare U as the stationary partner to the dynamical mass flux, M, of biological matter or chemical components and which is itself measured in kilogrammes per second across any t over T. Our U is then the kilogrammes maintained at that given t. We can then measure dU, the infinitesimal change in a population’s mass of components maintained at any t over T.
 We can track the population’s energy usage through V, its visible presence or energy density measured in kilogrammes per joule. We need only apply it to U and we immediately have both joules, H, and the Wallace pressure, P, in watts.
 N is the population count expressed in biomoles as already fully described.
 The four terms
Perhaps the biggest deficiencies, from our point of view, in biology and ecology are four:
 the failure to distinguish individual variables from population ones;
 the failure to distinguish mechanical chemical energy from nonmechanical chemical energy;
 the failure to distinguish exact from nonexact differentials in so far as this involves separating work done and heat emitted upon paths from the mass and energy needed to maintain the states from which those paths arise;
 the failure to establish a specified time period over which natural selection, competition and evolution can manifest themselves.
We shall attend to deficiency (d) shortly. In order to overcome deficiency (c) we need only carefully measure the paths and the actions over which biological activity occurs, and through which natural selection is therefore expressed. We achieve this by measuring all energy and by taking care where it is allocated … and which we very carefully did in our Brassica rapa experiment.
It is clear, regarding (a) and (b) from our list of requirements, that four partial differential terms must populate our desired equation. They must intersect so that regarding deficiency (a) above, two refer to the population at large; while two refer to the individuals within that population. And then in regard to deficiency (b), two must also refer to the assembled collection of whatever chemical components are currently being used, while two instead refer to the energy derived from those components. And since a biological population is at any given time t both upon a path, of length T, and also in a given state, then all variables will have both their dynamic and their stationary values for all ts over T. The dynamic and the stationary will automatically separate themselves into the exact and the inexact, with the dynamic then referring to the paths while the stationary refer to the given states at the ts upon those paths, and again for all ts over T. We have then corrected three of the above deficiencies … and we will be very close to turning biology and ecology into exact sciences comparable with any other, needing only to attend to (d) to complete the task.
With our five variables now selected; and with our terms and our purposes clarified; we next need a simplest possible case so the relevant partial differentials can be determined. And … nothing could be simpler than a population of two entities, A and B (Encyclopaedia Britannic, 2002). We want to speak of A and B together as a population, while still being able to refer to each one separately. This is mathematically straight forward. The advantage of a vector calculus based model such as we have derived is that partial derivatives are easily introduced and handled. It is the specific purpose of partial derivatives to hold all other variables constant while one is allowed to vary.
We first deal with the dynamic variables. These are M and P, the mass and the energy fluxes. Let our two entities have the initial masses mA and mB, and the initial Wallace pressures pA and pB. The totals for their mass and energy fluxes are M = mA + mB and P = pA + pB respectively. All relevant rates of change—dmA/dt, dmB/dt, dpA/dt, dpB/dt, dM/dt and dP/dt—can now be computed. There is also an average individual rate of change for mass and energy—dm̅/dt and dp̅/dt—over the entire population (of two!) and at every point. We also recollect that m̅ and p̅ are the divergences for the respective fluxes.
But we must also consider the entire generation time period, T, over which progeny is produced. We then also have the dynamical values M’, P’, m̅’ and p̅’ as generational averages for the fluxes and divergences. But these can now and immediately serve double duty as both (a) the weighted generational averages for the entire generation; and as (b) our vector unit normals and to which all other values, over the generation, can be referred and so which act as standards of measure … and with particular reference to an m^ and a p^ for the BiotSavart law. We therefore now have values for all our fluxes, divergences, and curls, and our relevant unit vector normals are in place to measure all values should that become necessary (which it will).
The above are all dynamic. They refer only to paths. Each such dynamic or path value can only exist because it is defined by a set of stationary states, with state variables, and as define that path through their moles of components and kilogrammes of mass and joules of energy and so forth. Every given dynamic Y has its stationary X where X = Y and with Y being stated per a time interval; and also with an x̅ and a y̅ being stated over the n in the population; and then with an X’, Y’, x̅’, y̅’ and so forth relative to the generation length, T.
Our population of two entities A and B therefore now also have the initial stationary values of uA and uB for their masses, and hA and hB for their chemical bond energies, with their totals being U = uA + uB and H = hA + hB. There are also and of course the respective rates of change—duA/dt, duB/dt, dhA/dt, dhB/dt, dU/dt and dH/dt, along with the population averages of du̅/dt and dh̅/dt again at every point. And since these values are held at each t over the same T as defines the path, then we also have a matching U’, H’, u̅’ and h̅’ as generational averages … with an u^ and an h^ being unit vector normals and standards of measure applicable to every t over T.
Two of our desired partial differentials must now refer to each of the two population fluxes, M and P. We shall call our two population partial differentials, when considered together, the population’s “essential development”, λ. We shall then call the partial differential that refers to the population mass, M, the “probundance”, γ; and we shall call the one that refers to the population’s energy, P, the “procreativity”, ψ. This then gives λ = γ + ψ for the population’s essential development. Its probundance, γ, is a quantity statable in moles—and therefore a specified mass—of chemical components; with its procreativity of energy, ψ, being derived from those components, and being expressible in joules and originating entirely from the chemical configurations applied. We can thus distinguish the mechanical from the nonmechanical for the population at large, and at any time, over the generation. We can also always distinguish exact differentials from nonexact ones.
And then similarly and complementarily, two of our partial differentials will refer to the two combined fluxes of M and P for the individuals within the population. We shall call these the “compensatory development”, L. We shall then call the partial differential that handles the individuals’ mass their “abundance”, C; and we shall call the partial differential that handles their energies their “accreativity”, Y. This then gives L = C + Y for the compensatory development applicable to the assorted individuals, and with respect to their abundance in amount of substance and so in mass of chemical components retained, C; and also in their accreativity of energy they derive from those specified components, Y, and according to the manner in which they configure the said components. We can therefore and again distinguish both the mechanical from nonmechanical, and the exact from the inexact over the various individuals in any population, and for any time t over T. If there are any other differentials we need we can soon find them which is the clear advantage of a good model.
 The equation
With these variables; a complete population; a list of requirements; and a clear list of our terms and their purposes in place; our Euler equation follows immediately:
µ 
= 
dS 
= 
(∂S/∂U)V,{Ni} dU 
+ 
(∂S/∂V)U,{Ni} dV 
+ 
Σi(∂S/∂ui)U,V,{Nj=/i} dui 
+ 
Σi(∂S/∂vi)U,V,{Nj=/i} dvi. 

1. Probundance, γ 
2. Procreativity, ψ 
3. Abundance, C 
4. Accreativity, Y 

Essential development, λ 
Compensatory development, L 
This equation now allows us to quantify µ which is natural selection and the specific effect of biological potential and biological activity causing any changes of any kind in the physiology and metabolism of all entities. We can isolate essential development, λ, which is those changes that apply only to the population as a whole, from any compensatory development, L, being those that apply specifically to individual entities:
 The probundance, γ. This is, again, the first part of essential development, λ, and speaks to how engeny, S, alters with respect to changes in the quantity of biological matter, U, when visible presence, V (i.e. the quantity of biological matter constructed per each unit of energy), and—most importantly—population numbers, N (in biomoles), are held constant where N measures how those components are apportioned into different entities. This probundance is therefore all changes in biological matter, U, being the total number of chemical components held in the population and also being the stationary correlate of the vector mass flux, M. But since N is being held constant, then this probundance also states the change in du̅, which is the average individual change in the numbers of chemical components held per entity. It therefore measures the individual changes in state that lead to the work done and heat emitted for that entire population. It is also therefore changes in m̅, which is the divergence of the mass flux. But in being a change in m̅ it is also a potential change in the vector curl, and so in the F responsible for the circulation of mass all around the boundary, T, and so over the population at large … and for the entire generation or any selected time interval, ΔT.
 The procreativity, ψ. This is the second part of the essential development, λ, and it speaks to how engeny, S, changes as the visible presence, V, changes while biological matter, U, and population numbers, N, hold constant. This is therefore all changes in the energy flux, P, that are not already being caused by changes in the mass of components held. But since the numbers, N, are again being held constant, then these are also changes in the divergence in the energy flux which is the average individual Wallace pressure, p̅, over the population. And again since N is constant; and also since H, the chemical bond energy over the population is a state variable; then this procreativity states the average individual change in dh̅ over all members and that accompanies any infinitesimal change dU, and so states any change in configuration in the entities. But in being a change in h̅, it is also of course a change in p̅, which is its dynamical correlate. This is therefore and again both a change in the divergence of energy, and also in its curl … but always as they affect entire populations and again for the whole generation, or else for any selected time interval ΔT.
 The abundance, C. This is the first part of our compensatory development, L. It represents all contributions to the change in engeny, S, with respect to changes in the components individually held by each entity as each entity and/or component theoreof is individually introduced into or else departs from the population … and as the population’s store of biological matter, U, visible presence, V, and overall population numbers, N, remain constant. This is therefore all changes in the divergence, and any associated with the curl, in the mass caused by any given entity by its insertion or removal. This is how we can begin to isolate and measure natural selection, competition, and evolution.
 The accreativity, Y. This is the second part of compensatory development, L. This fourth partial differential term represents all contributions to the engeny, S, induced by all changes in the individual chemical configurations or visible presences of each specified entity, but independent of its mass, and as each such entity is independently either introduced into, or else departs from, the system; and again as the population’s store of biological matter, U, visible presence, V, and numbers, N, remain constant. This is therefore both the divergence and the curl in energy caused by any given entity, and as it then affects the total energy flux and so work done or heat given off by the population and as it is inserted or removed.
This population equation gives us a complete accounting of all population variables and also of all variables for every entity within it, and as they all change and/or enter and/or are removed. And since we can now and in principle isolate essential development, λ, from compensatory development, L; and since each will have its separate portions of mechanical and nonmechanical energy; then we can also isolate all aspects of natural selection, competition, and evolution for they are together the biological potential. We will do so first for Brassica rapa, and then more generally for all species.