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
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