Information Technology Reference
In-Depth Information
Chapter 3
Systems of Ordinary Differential Equations
In Chap. 2, we saw that models of the form
y
0
.t /
D
F.y/;
y.0/
D
y
0
;
(3.1)
can be used to model natural processes. We observed that simple versions of (
3.1
)
can be solved analytically, and we saw that the problem can be solved adequately
by using numerical methods. The purpose of the present chapter is to extend our
knowledge to the case of systems of ordinary differential equations (ODEs), e.g.,
y
0
.t /
D
F.y;
z
/; y.0/
D
y
0
;
z
0
.t /
D
G.y;
z
/;
z
.0/
D
z
0
;
(3.2)
where y
0
and
z
0
are the given initial states and where F and G are smooth functions.
In Sect.
3.1
, we derive an interesting model of the form (
3.2
), which is of relevance
for the study of so-called predator-prey systems. Thereafter, we will discuss some
analytical aspects and study some numerical methods.
3.1
Rabbits and Foxes; Fish and Sharks
In Sect. 2.1.1 on page 32, we discussed models of population growth for a group of
rabbits on an isolated island. We first argued that the growth could be modeled by
y
0
D
˛y;
y.0/
D
y
0
;
(3.3)
where ˛>0denotes the growth rate and where y
0
is the initial number of rabbits
on the island. For relatively small populations we argued that (
3.3
) is a good model.
But as y increases - recall that the solution is y.t/
D
y
0
e
˛t
- the supply of food
will constrain the growth. To model the effect of limited resources, we introduced
the concept of carrying capacity ˇ and argued that
y
0
D
˛y.1
y=ˇ/;
y.0/
D
y
0
;
(3.4)
