18.03 Class 37
, May 7, 2010
Vector fields and nonlinear systems
1. Autonomous systems
2. Vector fields
3. Equilibria
4. Linearization
[1] Introduction to general nonlinear autonomous systems.
Recall that an ODE is autonomous if
x'
depends only on
x
and not on
t:
x' = f(x)
For example, I know an island in the St Lawrence River in upstate New York
where there are a lot of deer [Slide]. When there aren't many of them, they
multiply with growth rate
k ;
x' = kx .
Soon, though, they push up
against the limitations of the island; the growth rate is a function of
the population, and we might take it to be
k(1-(x/a))
where
a
is the
maximal sustainable population of deer on the island. So the equation is
x' = k(1-(x/a))x
,
the "logistic equation."
On this particular island,
k = 3
and
a = 3 ,
so
x' = (3-x)x .
(Maybe the units are hundreds of deer, and years).
There are "critical points" at
x = 0
and
x = 3 .
When
0 < x < 3 ,
x' > 0.
When
x > 3 ,
x' < 0 , and, unrealistically, when
x < 0 , x' < 0
too.
I drew some solutions, and then recalled the phase line:
------<-----*---->------*------<--------
One day, a wolf [Slide] swims across from the neighboring island, pulls himself
up the steep rocky coast, shakes the water off his fur, and sniffs the air.
Two wolves, actually.
Wolves eat deer, and this has a depressing effect on the growth rate of
deer. Let's model it by
x' = (3-x-y)x
where
y
measures the population of wolves.
Now, wolves in isolation follow a logistic equation too, say
y' = (1-y)y
(no deer)
But the presence of deer increases their growth rate, say
y' = (1-y+x)y