Predator-Prey and Competition Eqs in the Rare Case

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 09/29/04 

Date: Wed, 29 Sep 2004 14:17:27 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Predator-Prey and Competition Eqs in the Rare Case
Copyright By Owner Osher Doctorow Ph.D.
First Published 2004.

The Lotka-Volterra predator-prey system of equations was developed by
Lotka and Volterra in the 1920s. An excellent presentation of this
and other models and an introduction to applied mathematics is
Richard Haberman's (Rutgers U.) Mathematics Models Mechanical Vibra-
tions, Population Dynamics, and Traffic Flow, Prentice-Hall: Engle-
wood Cliffs, New Jersey 1977. I want to thank Chris Kappner for
showing me this excellent book in the late 1970s.

Both the Lotka-Volterra and Haberman's own Competing Populations
system become approximately linear near 0, as Haberman himself points
out. (0, 0) is an equilibrium point that is sometimes ignored
for population equations in the continuous versions because it is
a lower bound (Haberman himself describes it as not as important as
the nonzero equilibrium point for the Lotka-Volterra unlimited food
source case that he studies), but Haberman himself in his volume's
exercise sections explores the (0, 0) case. For continuity, (0, 0)
is best taken as the pair of proportions or percents or some
similarly real-valued pair of values on [0, 1] or [0, b] for b > 0.
The discrete gap between 0 and 1 for modellers who are thinking in
terms of population numbers (integers) is one of the sources of
the erroneous assumption that (0, 0) is not important.

The Lotka-Volterra equations are:

1) dF/dt = F(a - bF - cS)
   dS/dt = S(-k + LF)

with F the prey and S the predator variables (e.g. proportions) and
a, b, c, k, L nonnegative constants. Without food supply limitation
b is 0. However, even with food supply limitation, that is to say
nonzero b, notice that in [0, 1] the bF^2 term of the first equa-
tion of (1) can usually be taken as approximately 0 (for bF^2 < 1),
so the unlimited food supply case is very important:

2) dF/dt = aF - cFS
   dS/dt = -kS + LFS

and if S is not 0 we get the phase plane equation by dividing
dF/dt by dS/dt to get dF/dS:

3) dF/dS = F(a - cS)/[S(LF - k)]

which becomes singular ("0/0") at (0, 0) and (k/L, a/c)
which are pairs of equilibrium population (proportion) points.

Haberman's Competing Species equation system is (p. 247 ff):

4) dx/dt = x(a - bx - ky)
   dy/dt = y(c - dy - ox)

which near (0, 0) Haberman in his Exercises (p. 253) shows are
approximately:

5) dx/dt = ax
   dy/dt = cy

Equations (5) are separable so easily solved as exponentials in
time, and differ in the phase plane according to whether a > c,
a = c, or a < c.

Of course, (0, 0) and a small neighborhood of it are Rare Events
in probability terms (P(A) < .05) which can be extended to propor-
tions and so on. The Lotka-Volterra equations can be handled
similarly near (0, 0). Notice that for x in [0, 1) and y in
[0, 1), both xy and x^2 and y^2 are relatively small and quickly
become absolutely small (relatively compared to x and y) when
x and y are near 0.

Osher Doctorow