Biology Reference
In-Depth Information
x
0 and y
0, the system therefore has (x
0
,y
0
)
¼
(1,1) as its
only equilibrium state.
Next, we examine some analytic methods for determining the stability
of equilibrium points.
C. Some Necessary Mathematical Background
In order to determine the stability of the equilibrium point for a system
with two groups, we need to compute the partial derivatives of f (x,y)
and g(x,y).
In computing the partial derivative of, say:
x
2
y
3
f
ð
x
;
y
Þ¼
þ
3x
þ
4y
;
(2-7)
with respect to x, we treat y as if it were a constant and differentiate
the function as if x were the only variable. The symbol for the partial
derivative of f with respect to x is
@
f
x
. So, for Eq. (2-7):
@
@
f
2xy
3
x
¼
þ
3
:
@
Similarly, the partial derivative of f with respect to y is denoted by
@
f
y
.
@
For Eq. (2-7), this partial derivative is:
@
f
3x
2
y
2
y
¼
þ
4
:
@
Example 2-2
.......................
3xy. Find
@
f
x
and
@
f
2x
3
y
Let f (x,y)
¼
y
at the point (1,2).
@
@
S
OLUTION
:
We have:
@
f
6x
2
y
x
¼
3y
;
@
so that at the point (1,2):
@
f
1
2
x
ð
;
Þ¼
ð
Þð
Þ
ð
Þ¼
:
1
2
6
2
3
2
6
@
Likewise:
@
f
3x and
@
f
2x
3
1
3
y
¼
y
ð
1
;
2
Þ¼
2
ð
Þ
3
ð
1
Þ¼
1
:
@
@
