18.03 Class 8
, Feb 19, 2010
Autonomous equations
[1] Logistic equation
[2] Phase line
[3] Extrema, points of inflection
Announcements:
Final
Tuesday, May 18, 9:00  12:00, Johnson Track
Hour exam next Wednesday: Rooms to be announced Monday.
Please go to the hour you are registered for. 50 minutes.
[1] Back to qualitative study of differential equations.
I'll use
(t,y)
today.
The general first order equation is
y'(t) = F(t,y)
Autonomous ODE:
y'(t) = g(y) .
Autonomous means conditions are constant in time, though they may depend
on the current value of
y .
Eg [Natural growth/decay]
Constant growth rate:
so
y' = k0 y .
Eg Bank account with interest rate NOT depending on time but possibly
depending upon current balance, and constant savings rate:
y' = I(y) y + q .
Extended example: Population model with 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.
In the simplest version of this, when you graph of
k(y)
against
y
you
get a straight line with vertical intercept
k0
and horizontal intercept
p :
k(y) = k0 (1  (y/p)).
so
k(0) = k0 ,
and
k(p) = 0 . When
y > p ,
k(y) < 0 .
The Logistic Equation is
y' = k0 (1  (y/p)) y = g(y) .
This is more realistic than Natural Growth when you want to account for
limits to growth. It is nonlinear but it IS autonomoous.
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 Spring '09
 vogan
 Equations, Logistic function, stable population, phase line

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