Graphics Programs Reference
In-Depth Information
21.
Consideraclosedbiologicalsystempopulatedby
M
number of prey and
N
number of predators. Volterra postulated that the two populations are relatedby
the differentialequations
M
=
aM
−
bMN
N
=−
cN
+
dMN
where
a
,
b
,
c
and
d
areconstants. The steady-state solutionis
M
0
=
/
d
,
N
0
=
/
b
;
if numbers other than these are introducedinto the system, the populations
undergo periodic fluctuations. Introducing the notation
c
a
y
1
=
M
/
M
0
y
2
=
N
/
N
0
allows ustowrite the differentialequations as
y
1
=
a
(
y
1
−
y
1
y
2
)
y
2
=
b
(
−
y
2
+
y
1
y
2
)
Using
a
=
1
.
0
/
year,
b
=
0
.
2
/
year,
y
1
(0)
=
0
.
1 and
y
2
(0)
=
1
.
0, plot the two popu-
lationsfrom
t
=
0to50 years.
22.
The equations
u
=−
au
+
av
v
=
cu
−
v
−
uw
w
=−
bw
+
uv
known as the Lorenzequations, areencounteredintheory of fluid dynamics.
Letting
a
=
5
.
0,
b
=
0
.
9 and
c
=
8
.
2, solve these equationsfrom
t
=
0to10 with
the initialconditions
u
(0)
=
0,
v
(0)
=
1
.
0,
w
(0)
=
2
.
0 and plot
u
(
t
)
.
Repeat the
solutionwith
c
=
8
.
3
.
Whatconclusionscan youdrawfrom the results?
MATLAB Functions
[xSol,ySol] = ode23(dEqs,[xStart,xStop],yStart)
low-order (probably
third order) adaptive Runge-Kuttamethod. The function
dEqs
must return the
differentialequations as a column vector (recall that
runKut4
and
runKut5
require rowvectors). The rangeofintegrationisfrom
xStart
to
xStop
with the
initialconditions
yStart
(also a column vector).
[xSol,ySol] = ode45(dEqs,[xStart xStop],yStart)
issimilar to
ode23
, but
uses a higher-order Runge-Kuttamethod (probablyfifth order).
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