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for biology and ecology

μ = dS = (

∂S

∂U

)V,{Ni} dU + (

∂S

∂U

)U,{Ni} dV + Σi (

∂S

∂Ui

)U,V,{Nj≠i} dU + Σi (

∂S

∂Ui

)U,V,{Nj≠i} dV

µ is the population's “biological potential” in watts; *S* is the population's “engeny” in watts per kgs per unit population size, i.e. per “biomole”; Ut = ∫M dt, and is the quantity of biological matter held by the population through its mass flux over that given time interval in kgs; *V* is “visible presence”, the unit energy density and the inverse of the population's specific Gibbs energy in kgs per joule; N is entity number in the population expressed as a multiple of the unit population size, and so in “biomoles”.

m̅μ = m̅dS = dU + dH - Σiμi (
dvi - dmi )

*m̅* is the Lagrange multiplier for the population's energy … and also the population's divergence in its mass flux … and also the population's average individual mass flux in kgs per sec.; µ is the population's “biological potential” in watts; *S* is the population's “engeny” in watts per kgs per unit population size, i.e. per “biomole”;Ut = ∫M dt, and is the quantity of biological matter held by the population through its mass flux over the given time interval in kgs;Ht = ∫P dt, and is the population's total stock of energy held over the given time interval in joules; *v* is an individual entity's “visible presence” or unit energy density—i.e. that entity's inverse of its specific Gibbs energy, in kgs per joule; *m* is an individual entity's mass flux, or the mass it maintains at that moment in kgs. per sec.