From Osher Doctorow
The Lotka-Volterra or Predator-Prey equations:
1) dx/dt = -Ax(t) + Bxy
dy/dt = -Cy(t) + Dxy
with A, B, C, D positive constants reduce to exponential growth/decay
(expansion-contraction) if B and D are 0, but for the interpretation
of Bxy and Dxy which "look like" interaction terms, where x is the
predator population or proportion and y is the prey population or
proportion, we should really adopt a physics viewpoint and ask what xy
"looks like" to x and to y.
This brings us to some arguably very remarkable results if we use the
Probable Influence/Causation (PI) and Independent Probability-
Statistics viewpoints. Recall that the Riccati Differential equation
(which contains exponential growth/decay as a special case with the
square x or square y terms zeroed out) is the closest differential
equation to PI, while Independent Probability/Statistics (IPS) is
characterized by:
2) P(AB) = P(A)P(B) = xz when x = P(A), z = P(B) (in Independent
Probability-Statistics)
To (1) is a mixture of the PI and IPS scenarios. But from Fuzzy
Multivalued Logics and their Probability-Statistics analogs, the
General(ized) Boolean Logic Universe or the Probability-Statistics
Universe is divided into 3 scenarios: (a) PI or Lukaciewicz/Rational
Pavelka logical implications, (b) conditional probability or Product/
Goguen logical implications, (c) IPS or Godel logical implication.
As I explained in my recent Quantum Gravity thread sections on
sci.physics, PI only allows addition and subtraction and
multiplication by a constant k, so from the PI viewpoint, say that of
x or event/set A, xz is:
3) xz = kx (from viewpoint of x or A in PI, where k = P(B) = z)
So in the Lotka-Volterra equations, the predator or prey views its
opposite (prey or predator respectively) as constant and only itself
as "alive" or "changing". Thus, we do not have a real interaction in
a 2-way variable sense within the scenarios of PI or IPS. The view
from z is analogous to the view from x.
In the scenario of IPS, only the interaction (xz) is variable, while
the growth/decay of one of the populations is "constant", so that xz
is variable but neither x nor z alone is regarded as varying.
Although this defies "logic" outside Fuzzy Multivalued Logics, it is
not so surprising since for example human sociocultures often adopt a
"Pluralistic" viewpoint in which ony pluralities and not Individuals
(except for "Me" or "Number One", and sometimes not even that) are
recognized.
It is important to recognize that this idea extends deep into physics
and physical reality. That is to say, although physics undoubtedly
deals with the real world, nevertheless just as in SR and GR motion of
an object is relative to other objects, so the perception or view of
change in general depends on what scenario or phase or "Universe" in a
Multiple Universe one is in. It is to be hoped that there is a
"Super-Universe" in which everything is viewed accurately and in both
Individual and Plural terms (although Pluralities themselves are
nothing more than two or more (often large numbers of) Individuals).
However, that is a matter to be discovered, not assumed, and whether
we human beings have the capacity to be "Super-observers" remains to
be seen in view of the fundamental nature of the 3 different scenarios
in our Multiple Universes.
Osher Doctorow
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