# 42. A Gibbs-Duhem equation for natural selection

Since a sound mathematical model can be looked at in a variety of different ways, our interdependent intensive and extensive variables also allow us to present a biological version of a standard Gibbs-Duhem style equation:

 m̅µ = m̅dS = dU + dH - Σiµi(dvi - dmi) 1. Probundance, γ 2. Procreativity, ψ 3. Accreativity, Y; and 4. Abundance, C Essential development, λ Compensatory development, L

Both dU and dH again reflect the probundance, γ, and the procreativity, ψ, respectively, of essential development, λ, these being properties of the entire population rather than of any given entity and such that λ = γ + ψ. These are the changes in the population’s stock of biological matter, U, and energy, H, when the numbers hold constant … and so when µ = dS = dU + dH = λ = γ + ψ and the remaining summation term for compensatory development is zero.

We can now specifically consider, say, all variables and variations associated with the ith entity in any population. Its specific effects on our population are given by its specific biological potential or apportionment tendency regarding its own mass and energy:

µi = (U/mi)V,S,N - (U/vi)m_,S,N.

This states that if we hold the visible presence, the engeny, and the numbers over the rest of the population constant, and infinitesimally increment the number of components introduced because of i; and/or hold the average number of components per entity, the engeny, and the numbers for the population constant and then infinitesimally increase is visible presence by introducing it; then we will state all the energy changes caused by this ith entity … including its insertion into or extraction from any population. And … this is exactly the effect we need to quantify natural selection and Darwinian competition and evolution.

This biological potential, µ—and which we can measure in watts—is now the motive force for the biological cycle. It is responsible for any energy flux. It suitably partitions itself amongst its three different factors and to affect both the entire population and its individual entities. Therefore … the energy change inhering to every biological entity in any and all biological systems is measurable and determinable … as also is any change that could be conjectured under the constraint that N has remained constant and that all terms due to all distinct is or separate entities have been either ignored or considered. We can in other words test Darwinain competition against predictions made both with and without it; take measurements; compare the measurements to our predictions; and then draw appropriate conclusions.

We can now state simply and clearly that if competition, natural selection, and Darwinian evolution do not exist, then all terms declaring a compensatory development, L, in our above two equations will sum to zero. That is to say, the two terms Σi(S/ui)U,V,{Nj=/i} and dui, Σi(S/vi)U,V,{Nj=/i} dvi from our first equation, and Σiµi(dvi - dmi) from the second—being the abundance, C, and the accreativity, Y— must again all be zero. They are called the compensatory development, L, precisely because they are the biological inductions proposed by Darwin and that can only occur in response to the changes in numbers that he proposed, as members are lost to the environment and regained in reproduction, for those measurable biological inductions are an explicit effort to preserve the species and the population through compensatory changes in all remaining entities.