In the 1920 's, the Italian mathematician Umberto Volterra proposed the
following mathematical model of a predator-prey situation to explain why,
during the first World War, a larger percentage of the catch of Italian
fishermen consisted of sharks and other fish eating fish than was true both
before and after the war. Let \(x(t)\) denote the population of the prey, and
let \(y(t)\) denote the population of the predators.
In the absence of the predators, the prey population would have a birth rate
greater than its death rate, and consequently would grow according to the
exponential model of population growth, i.e. the growth rate of the population
would be proportional to the population itself. The presence of the predator
population has the effect of reducing the growth rate, and this reduction
depends on the number of encounters between individuals of the two species.
Since it is reasonable to assume that the number of such encounters is
proportional to the number of individuals of each population, the reduction in
the growth rate is also proportional to the product of the two populations,
i.e., there are constants \(a\) and \(b\) such that
$$
x^{\prime}=a x-b x y
$$
Since the predator population depends on the prey population for its food
supply it is natural to assume that in the absence of the prey population, the
predator population would actually decrease, i.e. the growth rate would be
negative. Furthermore the (negative) growth rate is proportional to the
population. The presence of the prey population would provide a source of
food, so it would increase the growth rate of the predator species. By the
same reasoning used for the prey species, this increase would be proportional
to the product of the two populations. Thus, there are constants \(c\) and \(d\)
such that
$$
y^{\prime}=-c y+d x y .
$$
a) A typical example would be with the constants given by \(a=0.4, b=0.01,
c=0.3\), and \(d=0.005\). Start with initial conditions \(x_{1}(0)=50\) and
\(x_{2}(0)=30\), and compute the solution to (8.22) and (8.23) over the interval
\([0,100]\). Prepare both a time plot and a phase plane plot.
After Volterra had obtained his model of the predator-prey populations, he
improved it to include the effect of "fishing," or more generally of a removal
of individuals of the two populations which does not discriminate between the
two species. The effect would be a reduction in the growth rate for each of
the populations by an amount which is proportional to the individual
populations. Furthermore, if the removal is truly indiscriminate, the
proportionality constant will be the same in each case. Thus, the model in
equations \((8.22)\) and (8.23) must be changed to
$$
\begin{aligned}
&x^{\prime}=a x-b x y-e x \\
&y^{\prime}=-c y+d x y-e y
\end{aligned}
$$
where \(e\) is another constant.
b) To see the effect of indiscriminate reduction, compute the solutions to the
system in \((8.24)\) when \(e=0\), \(0.01,0.02,0.03\), and \(0.04\), and the other
constants are the same as they were in part a). Plot the five solutions on the
same phase plane, and label them properly.
c) Can you use the plot you constructed in part b) to explain why the
fishermen caught more sharks during World War I? You can assume that because
of the war they did less fishing.
