Stable and Unstable Fixed Points
Those who are deadly against math must be bored to the point of
abondoning us. But the goal is now in sight, and we do the final
climbing of technical barrier.
We go back to the M = 2
species lotka-Volterra equation
dx1/dt =
b1 x1 + a11
x12 + a12 x1
x2
dx2/dt = b2
x2 + a21 x1 x2 +
a22 x22
Fixed point
(X1, X2) of this equation which is neither
X1 = 0 nor X2 = 0 (referred to as "nontrivial
fixed point") is given by
0 = b1 +
a11 X1 + a12 X2
0 = b2 + a21 X1 + a22
X2
Suupose x1(t),
x2(t) have the value close to this fixed point. We write
them as
x1 = X1 +
u
x2 + X2 + v
Insert these into
the Lotka-Volterra equation and neglecting squred terms (because
u(t), v(t) themselves are small, their squres must be negligible),
we obtain
du/dt = a11 X1 u +
a12 X1 v
dv/dt = a21
X2 u + a22 X2 v
This set of
equation describes a Linear Map, and depending
on the values of aij and
Xi , there are several
different types of solutions to it conserning long time (large t)
behavior. We list two important types for ou discussion.
(1)
attracting vortex : This occurs when the eigenvalues (ehem;) of the
linear maps are both negative
u(t) --> 0, v(t) --> 0
Decrease while oscillating.
(2) hyperbolic : This occurs
when the eigenvalues of the linear maps are both positive (I told
you that this chapter's rather techy)
u(t) -->∞, v(t)
--> ∞ diverge. All orbits get repulsed after approaching the
fixed point.
Lookingg the picture
of previous chapter, "prey-predator" is of type (1), and
"competitors" (2). |
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