452 Ordinary Differential Equations
10.5 Models for Interacting Species
We examined the Malthusian model and the logistic model (Verhulst-Pearl
model) for population growth of a single species in chapter 3. In this sec-
tion, we will examine several models which attempt to represent population
dynamics when two or more species interact in the same environment.
Volterra-Lotka Prey-Predator Model
The first model which we will study is named in honor of the American
scientist and statistician Alfred Lotka and the Italian mathematician Vito
Volterra. Both men studied this model at about the same time and arrived at
similar conclusions. This particular model serves as the cornerstone for the
study of population dynamics for interacting species.
On March 2, 1880, Alfred James Lotka (1880-1949) was born in Lemberg,
Austria, to American parents. He received his elementary and secondary
education in France. In 1901, he was granted a B.Sc. degree by Bingham
University in England. During 1901-2, Lotka pursued graduate studies at the
University of Leipzig in Germany. In 1902, he moved to the United States and
worked as a chemist for the General Chemical Company until 1908. During
1908-9 he was an assistant in physics at Cornell University. In 1909, Cornell
University granted him an M.A. degree. He worked briefly as an examiner
for the U.S. Patent Office and from 1909-11 he was a physicist for the U.S.
Bureau of Standards. In 1912, Lotka received his D.Sc. degree from Bingham
University. He then returned to work for the General Chemical Company
from 1914-19. In 1924, he joined the statistical bureau of the Metropolitan
Life Insurance Company in New York City. He remained there until his death
on December 5, 1949.
Alfred Lotka was the first person to systematically analyze the wide variety
of relationships which occur between two interacting species and to formulate
and study mathematical models to represent those interactions. Some of the
models which we are about to study appeared in his 1925 book, Elements of
Physical Biology.
The Italian mathematician Vito Volterra (1860-1940) independently con-
structed the same basic prey-predator population model as Lotka and arrived
at many of the same conclusions. Volterra was born on May 3, 1860, in An-
cona, Italy. He began studying geometry at age 11 and calculus at age 14.
In 1882, he received his doctorate in physics from the University of Pisa. His
first appointment was as professor of mechanics and mathematics at Pisa.
Later, he was a faculty member of the University of Rome for a period of
thirty years. Volterra's major contributions to mathematics are in the areas