Title: The paradox of Vito Volterra's predator-prey model,
Abstract: This article is dedicated to the late Giorgio Israel. R{é}sum{é}. The aim
of this article is to propose on the one hand a brief history of modeling
starting from the works of Fibonacci, Robert Malthus, Pierre Francis Verhulst
and then Vito Volterra and, on the other hand, to present the main hypotheses
of the very famous but very little known predator-prey model elaborated in the
1920s by Volterra in order to solve a problem posed by his son-in-law, Umberto
D'Ancona. It is thus shown that, contrary to a widely-held notion, Volterra's
model is realistic and his seminal work laid the groundwork for modern
population dynamics and mathematical ecology, including seasonality, migration,
pollution and more. 1. A short history of modeling 1.1. The Malthusian model.
If the rst scientic view of population growth seems to be that of Leonardo
Fibonacci [2], also called Leonardo of Pisa, whose famous sequence of numbers
was presented in his Liber abaci (1202) as a solution to a population growth
problem, the modern foundations of population dynamics clearly date from Thomas
Robert Malthus [20]. Considering an ideal population consisting of a single
homogeneous animal species, that is, neglecting the variations in age, size and
any periodicity for birth or mortality, and which lives alone in an invariable
environment or coexists with other species without any direct or indirect
inuence, he founded in 1798, with his celebrated claim Population, when
unchecked, increases in a geometrical ratio, the paradigm of exponential
growth. This consists in assuming that the increase of the number N (t) of
individuals of this population, during a short interval of time, is
proportional to N (t). This translates to the following dierential equation :
(1) dN (t) dt = $\epsilon$N (t) where $\epsilon$ is a constant factor of
proportionality that represents the growth coe-cient or growth rate. By
integrating (1) we obtain the law of exponential growth or law of Malthusian
growth (see Fig. 1). This law, which does not take into account the limits
imposed by the environment on growth and which is in disagreement with the
actual facts, had a profound inuence on Charles Darwin's work on natural
selection. Indeed, Darwin [1] founded the idea of survival of the ttest on the
1. According to Frontier and Pichod-Viale [3] the correct terminology should be
population kinetics, since the interaction between species cannot be
represented by forces. 2. A population is dened as the set of individuals of
the same species living on the same territory and able to reproduce among
themselves.