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18.03 Class 8, Feb 24, 2006
Autonomous equations
I'll use
(t,y)
today.
y' = F(t,y)
is the general first order equation
Autonomous ODE:
y' = g(y) .
Eg [Natural growth/decay]
Constant growth rate:
so
y' = k0 y .
k0 > 0
means the populuation (if positive) is growing;
k0 < 0 means it
is falling, decaying.
Autonomous means conditions are constant in time, though they may depend
on the current value of
y .
Eg [Logistic equation]
Variable growth rate
k(y), depending on the
current population but NOT ON TIME; so
y' = k(y) y.
Suppose that when
y
is small the growth rate is approximately
k0 ,
but that there is a maximal sustainable population
p , and as
y
gets near to
p the growth rate decreases to zero. When
y > p , the growth rate becomes
negative; the population declines back to the maximal sustainable
population.
Simplest version:
A graph of
k(y)
against
y , straight line with vertical intercept
k0 and horizontal intercept
p :
k(y) = k0 (1  (y/p)).
so
k(0) = k0 ,
and
k(p) = 0 .
The Logistic Equation is
y' = k0 (1  (y/p)) y = g(y) .
This is more realistic than Nat Growth for large populations. It is
nonlinear.
Autonomous equations are always separable, but we aim for a
qualitative grasp of solutions. Sketch direction field, isoclines first.
Values of
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 Winter '08
 Staff
 Equations

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