Biomedical Engineering Reference
In-Depth Information
Figure 7.26
Time evolution of populations of preys and predators.
The simplest—but very interesting—model is that of Lotka-Volterra. If
A
and
B
represent the populations of preys and predators, their time evolution is given by
[9, 10]
¶
A
=
aA bAB
-
¶
¶
t
B
(7.47)
= -
cB d AB
+
¶
t
In (7.47) the term
aA
represents the growth of population
A
if predators were
absent—
a
being the rate of birth, and the term
bAB
the decrease in the number of
prey due to the action of the predators (for these reasons, it is proportional to
A
and
to
B
). On the other hand, the term
cB
represents the mortality rate of predators (
b
being the rate of deaths) and the term
dAB
the prey contribution (as a source term)
to the predator growth rate (proportional to
A
and
B
).
Mathematically speaking, the system (7.47) is strongly coupled and nonlinear.
It is also structurally not stable. However, it bears much of the physics of the evolu-
tion of the prey-predator system. A first step in analyzing the Lotka-Volterra model
is to render the system nondimensional by introducing the new parameters
c
τ
=
at
;
α
=
a
(7.48)
A
B
u d
v
b
=
;
=
c
a
Substituting (7.48) in (7.47) yields
¶
u
=
u
(1
-
v
)
¶
¶
τ
(7.49)
v
=
α
v u
(
-
1)
¶
τ