The population equations
for biology and ecology

The Euler equation for biology

μ = dS = (∂S/∂U)_{V,N_i} dU + (∂S/∂V)_{U,N_i} dV + Σ_{_i} (∂S/∂u_i)_{U,V,{N_j≠i}} du_i + Σ_{_i} (∂S/∂v_i)_{U,V,{N_j≠i}} dv_i

µ is the population's “biological potential” in watts; S is the population's “engeny” in watts per kgs per “biomole”; U = ∫M dt, and is the quantity of biological matter in the population's mass flux at that moment, t, in kgs; V is “visible presence”, the inverse of the population's specific Gibbs energy in kgs per joule; N is entity number in the population in “biomoles”.

Explanation

The Gibbs-Duhem equation for biology

m̅μ = m̅dS = dU + dH - Σ_{_i} μ_i(dv_i - dm_i)

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 poppulation's “engeny” in watts per kgs per “biomole”; U = ∫M dt, is the quantity of biological matter in the population's mass flux at that moment, t, in kgs; H = ∫P dt, the population's total energy flux at that moment, t, in joules; v is an individual entity's “visible presence”, 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 in kgs. per sec.

Explanation

All biological organisms are subject to the three constraints of constant propagation, P > 0 = ∫₀^T dP, constant size, R > 0 = ∫₀^T dR, and constant equivalence, W > 0 = ∫₀^T dW.

Further information

Biology is “the study of those thermodynamic systems that can replace their internal energy”.

dU = Mdt = δQ - dH

Ecology is “the study of the exact processes by which a thermodynamic system replaces its internal energy”.

pdt + mdt = dh + du

G O T O P A P E R

The four laws of biology

LAW 1: The law of existence
n >= 1; δW = (δQ - dU) > 0; m → ∞; m̅ > 0

LAW 2: The law of equivalence
(δW₁ = δW₂) ∧ (δW₂ = δW₃) ⇒ (δW₁ = δW₃)

LAW 3: The law of diversity
A ⇒ 0; F ⇒ M

LAW 4: The law of reproduction
dA/dt > 0; dm̅/dt < 0; m̅ > 0; dn/dt >= 0

The four maxims of ecology

MAXIM 1: The maxim of dissipation
[Darwin's theory of competition]
∫dm < 0; ∇• M → 0; M = nm̅

MAXIM 2: The maxim of number
∇• H = δW = Pdt = nh̅

MAXIM 3: The maxim of succession
[Darwin's theory of evolution]
∇ x M = ∂m̅/∂t - ∂n/∂t

MAXIM 4: The maxim of apportionment
∇ x H = ∂m̅/∂t - ∂n/∂t - ∂V/∂t

www.engenetics.net

The population equationsfor biology and ecology

The Euler equation for biology

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

Explanation

The Gibbs-Duhem equation for biology

m̅μ = m̅dS = dU + dH - Σi μi(dvi - dmi)

Explanation

All biological organisms are subject to the three constraints of constant propagation, P > 0 = ∫0T dP, constant size, R > 0 = ∫0T dR, and constant equivalence, W > 0 = ∫0T dW.

Further information

Biology is “the study of those thermodynamic systems that can replace their internal energy”.

dU = Mdt = δQ - dH

Ecology is “the study of the exact processes by which a thermodynamic system replaces its internal energy”.

pdt + mdt = dh + du

The population equations
for biology and ecology

μ = dS = (∂S/∂U)_{V,N_i} dU + (∂S/∂V)_{U,N_i} dV + Σ_{_i} (∂S/∂u_i)_{U,V,{N_j≠i}} du_i + Σ_{_i} (∂S/∂v_i)_{U,V,{N_j≠i}} dv_i

m̅μ = m̅dS = dU + dH - Σ_{_i} μ_i(dv_i - dm_i)

All biological organisms are subject to the three constraints of constant propagation, P > 0 = ∫₀^T dP, constant size, R > 0 = ∫₀^T dR, and constant equivalence, W > 0 = ∫₀^T dW.