Re: The Riccati Matrix Exponential and Systems of Equations
From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/30/04
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Date: Mon, 30 Aug 2004 11:48:51 +0000 (UTC)
On 30 Aug 04 00:39:11 -0400 (EDT), Osher Doctorow wrote:
>for y = (u, v)^T (T meaning transpose), we get surprisingly close
>to the brain neuron equations which I have just discussed under
>another thread, at least for some of the equations. In a way, this
>is not surprising since linearizations of nonlinear systems hold
>in small neighborhoods at least. But we do have the problem of
>what to do if one u^2 instead of u occurs, as for example dv/dt =
>a21u^2 + a22v, du/dt = a11u + a23v. Likewise, if uv replaces one
>of the u or v quantities, what do we do? The Lotka-Volterra and
>similar equations also have this difficulty. I will try to discuss
The Lotka-Volterra predator-prey equations without a quadratic form
are written:
1) dx/dt = ax - bxy
2) dy/dt = -cy + pxy
where x is prey, y is predator. Look what happens when we calculate
d(x - y)/dt from these:
3) d(x - y)/dt = ax - bxy + cy - pxy = ax + cy -(b + p)xy
Readers who have followed my contributions will notice the Jacobson
Radical star product:
4) x o y = x + y - xy
except that the product is weighted by a, c, and b + p. In other
words:
5) (ax o cy)_(bp) = ax + cy - (b + p)xy
where _(bp) is a subscript indicating that b + p multiplies the -xy
term.
Richard Haberman (1977) gives the Lotka-Volterra version:
6) dF/dt = F(a - bF - cS)
7) dS/dt = S(-k + LF)
with a, b, c, k, L constants, where F is population of fish (prey),
S is population of sharks (predators). We can calculate:
8) d(F - S)/dt = aF + kS - (c + L)FS - bF^2 = (aF o kS)_(cL) - bF^2
Haberman's volume, Mathematical Models Mechanical Vibrations, Popula-
tion Dynamics, and Traffic Flow, Prentice-Hall: Englewood Cliffs,
New Jersey, 1977, linearizes (6) and (7), but then finds that in
some cases the equilibrium population is on the borderline between
stability and instability, so he directly studies the nonlinear
equations by defining an auxiliary variable Z:
9) Z = F^(-k)exp(LF)
which turns out to equal E_0 S^a exp(-cS) upon exponentiating the
equation for dF/dS upon eliminating time t by dF/dS = (dF/dt)/(dS/dt)
and the population of fish and sharks fluctuate periodically about
their equilibrium values in what look like two somewhat out of phase
sine or cosine curves with the fish leading the sharks. Since
almost any equation(s) can be made periodic by a trick (as I
described in fairly recent postings), I suggest that neither the
fish nor the shark population is relevant but rather their diff-
erence, and according to equation (8) the fish population F is
pulling down the difference F - S in its time variation proportion-
ately to F^2 from the "optimally large" star product (recall that
in the Jacobson Radical, the star product is counter-optimal as it
approaches 0 from above for real variables).
One difference between the Lotka-Volterra equations and the various
neuron models of the brain (see my previous thread) is that popula-
tions of discrete individuals are highly artificial from a biolog-
ical viewpoint. It is similar to decision-making based on numbers
of people: in PI, the fewer the numbers of people in a nation, the
more influential is the nation under quite general conditions (see
another of my previous threads).
Osher Doctorow
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