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# 39. The fourth maxim of ecology: the maxim of apportionment

The logic for the final maxim of ecology is surely obvious. It must be the curl or vector cross-product for the energy flux, and based on the divergence in energy, h̅ = H/n, with H being the stationary equivalent of the dynamic Wallace pressure, P.

- Maxim 4: Proviso 1

If the divergence in mass increases then m̅, the average individual mass, increases … and the average individual energy, p̅, also increases for the mechanical chemical energy needed to form the chemical bonds needed to bind those components must increase. The energy flux therefore depends upon changes in the mass flux, so we again have:

- +∂m̅/∂t

- Maxim 4: Proviso 2

The population count for energy is taken over exactly the same entities as for the mass flux, and so therefore if the population count decreases then the divergences and the curls of both the mass and the energy must increase together and for exactly the same reasons, they have the identical population count. So we again have:

- -∂n/∂t.

- Maxim 4: Proviso 3

The energy flux has, however, one further variable. It is entirely permissible for DNA to for example fold or unfold proteins. The energy stock changes while the population’s mass of chemical components remains the same. A population’s stock of biological energy is therefore apportioned across (a) its mass of components held per second, M, which is its mechanical chemical energy; (b) its numbers, n; and (c) its nonmechanical energy which is its Gibbs energy per unit mass, its energy density and visible presence, V.

- If a population’s energy flux increases because its DNA reconfigures its mass of chemical components, then the curl or circulation of energy per unit area immediately increases and the visible presence—which is the invert of energy—moves oppositely and decreases.
- If DNA is oppositely reconfigured so the net energy flux decreases, then the curl decreases as does the net circulation, while the visible presence increases.
- If the population’s energy holds constant while the number of components decreases then the visible presence decreases and the curl increases.
- If the population’s energy holds constant while the number of components increases, then visible presence increases and curl decreases for the energy can now circulate in a more leisurely fashion over an increased surface area.

Visible presence contributes to the unit energy circulation or energy density through its command over how biological potentials and components are configured, which is the nonmechanical energy and the vector curl. And since it is an inverse and moves oppositely to changes in energy, we therefore have:

- -∂V/∂t

Since the biological cycle cannot proceed without ∂V/∂t or changes in visible presence, then all such changes are now formally called “essential development”, λ. Essential development is simply all biological changes in state as are not caused either by (a) changes in mass, or by (b) changes in numbers. Therefore, the appropriate vector cross-product for all changes in the energy flux, which are changes in how energy moves through the population, as a volume, and across our Gaussian surface is given by:

The Fourth Maxim of Ecology: The Maxim of Apportionment

∇ x H = ∂m̅/∂t - ∂n/∂t - ∂V/∂t

The bioactivity of a biological population is subject to increase from an initial value for one or more of three reasons: (a) increases in mass; (b) decreases in competition. All other increases are due to (c) the essential development of the entity or species.