The three constraints

CONSTRAINT 1: The constraint of constant propagation

0 =  T0dP < P’ = N’p̅’

Every biological population is associated with a collection of traits, behaviours, and cultural artefacts and information that change constantly, but that can always be divided into discrete elements all of which have the potential to be mimicked and transferred from one individual entity within the population to another.

P is an extensive variable and is the population's “Wallace pressure”—its total energy flux and output in joules per second.


CONSTRAINT 2: The constraint of constant size

0 =  T0dM < M’ = N’m̅’ ( = R’)

The individual entities in every biological population are genotypes which together encode that population’s genome or collective genetic encoding, which at least some amongst them are able to reproduce and recreate.

M is an extensive variable, and is the total mass of chemical components the population holds at any time; R is an intensive variable and is the population's average energy content expressed in “darwins” or joules per “biomole” (where the biomole is a unit of population size).


CONSTRAINT 3: The constraint of constant equivalence

0 =  T0dS < S’ = N’s̅’

[A] No individual biological entity can be separated from all possible discrete elements of traits, behaviours, and cultural artefacts and information.

(Corollary: prodigious savant are always possible; and even “the walls of rude minds are scrawled all over with facts, with thoughts” Ralph Waldo Emerson).

[B] Not all the discrete elements associated with any one trait, behaviour, or cultural artefact or information can be uniquely attributed to any one biological entity.

(Corollary: “Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than them, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size” John of Salisbury, Metalogicon, 1159).

S is an intensive variable and is the population's (specific) work rate or its dynamical specific energy, but expressed as watts delivered per kilogrammes of mass maintained per “biomole”.


The Euler equation for biology, governing all biological activity

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

The Gibbs-Duhem equation for biology, governing all biological energy

m̅μ = dS = dU + dH - Σi μi(dvi - dmi)

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

dU = Mdt = δQ - dH; M > 0 (See explanation of terms and variables)

Ecology is “the study of the processes systems use to replace their internal energy”.

pdt + mdt = dh + du = δq; m > 0 (See explanation of terms and variables)


G  O    T  O    E  Q  U  A  T  I  O  N  S

The four laws of biology

brassica rapa experiment

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

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

LAW 3: The law of diversity
A → 0; FM

LAW 4: The law of reproduction
[(dm̅/dt ≤ 0) ∧ ( > 0)] ⇒ [(dn/dt ≥ 0) ∧ (dA/dt > 0)]

The four maxims of ecology

brassica rapa experiment

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

MAXIM 2: The maxim of number
∇ • H = H/n =

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

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