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is a better model. We saw, both analytically and numerically, that this model gives
predictions that are consistent with our intuition. Data that really support logistic
solutions, i.e., solutions of (
3.4
), can be found in [5].
In this chapter we will introduce a predator-prey model. Imagine that we intro-
duce a group of foxes to the isolated island. The foxes (predators) will eat rabbits
(prey) and a decrease in the number of rabbits will result in a decrease in the food
resources for the foxes. This will in turn result in a decrease in the number of foxes.
Our aim is to model this interaction. A similar situation appears if one studies the
growth of fish (prey) and sharks (predators). The model that we will develop is
usually referred to as the Lotka-Volterra model.
1
The background of Volterra's interest in this topic was that fishermen observed
a somewhat strange situation immediately after World War I. Fishing was more
or less impossible in the upper Adriatic Sea during World War I. After the war,
the fishermen expected very rich resources of fish. But contrary to their intuition,
they found very little fish. This fact was surprising and Volterra tried to develop a
mathematical model that could shed some light on the mystery. He derived a 2
2
system of ODEs of the form (
3.2
). Volterra's model is now classic in mathematical
ecology and it demonstrates the basic features of predator-prey systems.
We will follow Volterra's
2
derivation of a model for the population density of fish
and sharks, but keep in mind that exactly the same line of argument is valid for the
rabbit-fox system.
Let F
D
F.t/ denote the number of fish located in a specific region of the sea at
time t . Similarly, S
D
S.t/ denotes the number of sharks in the same region. Let us
first consider the case of S
D
0, i.e., no sharks are present. Then, the growth of fish
can be modeled by
F
0
D
˛F ;
(3.5)
which predicts exponential growth. Here ˛>0is the growth rate. As we observed
above, limited resources lead to a logistic model of the form
F
0
D
˛F .1
F=ˇ/;
(3.6)
where ˇ>0is the carrying capacity of the environment. This equation can be
rewritten as
F
0
F
D
˛.1
F=ˇ/ ;
(3.7)
1
Vito Volterra (1860-1940) was an Italian mathematician who worked on functional analysis,
integral equations, partial differential equations and mathematical models of biology.
2
Our presentation is based on the classic topic of Richard Haberman [17]. We strongly recommend
the reader to read more about population growth in that book or in the topic of M. Braun [7], which
is also heavily used as background material for the present text.
