1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 453

of functional analysis and integral equations. The following is an account of
the events which led Volterra to his formulation of the prey-predator model.
From 1910 to 1923 , Humberto D 'Ancona collected data on the number
of each species of fish sold in the markets of Trieste, Fiume, and Venice.
D 'Ancona assumed that the relative numbers of the various species available
at the markets indicated the relative numbers of the species in the Adriatic
Sea. He noticed that the percentage of predator fish (sharks, skates, rays,
etc.) in the total fish population was higher during and immediately after
World War I (1914-18). He concluded that the reduction in fishing due to
the war caused the change in the ratio of predators to prey. He reasoned
that during the war the ratio was close to its natural state and that the
decrease in this ratio before and after the war was due to fishing. D 'Ancona
hypothesized that the reduction in the level of fishing due to the war caused
an increase in the number of prey fish which in turn caused a larger increase
in the number of predator fish- thus, accounting for the increased percentage
of predators. However, he could not give biological or ecological reasons why
fishing should be more beneficial to prey than to their predators. D 'Ancona
requested his father-in-law, Vito Volterra, to construct some mathematical
models to explain the situation. In a few months, Volterra formulated a set
of models for the population dynamics for two or more interacting species.

The following sequence of assumptions and reasoning may be similar to
those which led both Lotka and Volterra to their elementary prey-predator
model. Let x(t) be the population (number) of prey at time t and let y(t) be
the population of predators at time t.

Assumption 1 The rates of change of the populations depends only on x
and y.

This assumption means the prey-predator population dynamics can be mod-
ell ed by an autonomous system of differential equations of the form

dx

dt = f(x, y)

(1)

dy

dt = g(x, y).

Assumption 2 The functions f and g are quadratic in x and y.

This assumption means the functions f and g have the following forms

(2) f(x, y) = A+ Bx+ Cx^2 +Dy+ Ey^2 + Hxy

and

(3) g(x,y) = K+Lx+Mx^2 +Ny+Py^2 +Qxy

where A, B, C, D , E, H, K , L , M, N, P, and Qare all real constants.