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Introduction
During his childhood, Andrei Nikolaevich Kolmogorov found biology very interesting. In fact, in the book Kolmogorov in Perspective, one can read that he made the following comment about himself as a schoolboy:

“I was one of the best in my class in mathematics, but my real scientific passions were, first of all biology, and then russian history” ([Kol00], p. 5)

He kept these centers of interest for the rest of his life. Thus, in 1940, Kolmogorov dared to confront the feared Lysenko by defending Mendel’s laws − a very dangerous move to make in the middle of Stalin’s regime. And, according to V. I. Arnold, the last research done by Kolmogorov [KB67], published in 1967, was motivated by biological ideas about the structure of the brain ([Kol00], p. 94).

In fact, Kolmogorov made only a few isolated contributions to biomathematics; but they all demonstrate, as one would expect, a remarkable originality. In particular, the short note [Kol36] about the predator-prey equation is a model of perspicacity and has had great influence on the deterministic theory of population dynamics. It is one of the rare articles that Kolmogorov published in Italian, doubtlessly in honor of the mathematician Vito Volterra who inaugurated what would later be called The Golden Age of biomathematics [SZ78]. Kolmogorov’s note represents a qualitative jump in the theory of predator-prey systems.

From Volterra equations to Gause equations
The beginning point of the study of Kolmogorov discussed here is the famous model that Volterra1 used, as early as 1925, to explain a surprising discovery of his son in law, the ecologist Umberto D’Ancona [Kin85]. Because of his research in marine biology, based on statistics from fish markets, D’Ancona noticed that during World War 1, the number of predators among Adriatic fauna had increased while the number of prey had diminished. This seemed to be an effect of the reduction of fishing due to the Austro-Italian hostilities: but why did it work in this manner and not in another?

Volterra based his argument on an ordinary differential equation: if $$x(t)$$ and $$y(t)$$ are the densities of prey and of predators, respectively, then the rate of increase $$\dot{x}/x$$ of the prey should be a decreasing function of y, positive for $$y=0$$, and the rate of increase $$\dot{y}/y$$ an increasing function of x, negative for $$x=0$$. If we suppose that these functions are linear, we see that

$$\dot{x} = x\bigl(a-by\bigr)$$

$$\dot{y} = y\bigl(-c+dx\bigr)$$

where the constants a, b, c, d are positive. In the positive quadrant, the phase portrait consists of periodic orbits around the equilibrium position (¯x, y¯)=(c/d, a/b). Volterra showed that the temporal averages of x(t) and y(t) along periodic orbits coincide with the values $$\bar{x}$$ and $$\bar{y}$$, which gave him a way to explain D’Ancona’s observation: in fact, the supplementary contribution due to the fishermen’s work diminishes the quantity a (the rate of increase of the prey in the absence of predators) and increases c (the rate of decrease of predators in the absence of prey), without affecting the values of the coefficients b and d, which measure the effects of the interaction between the predators and their prey. The corresponding effect on the temporal averages of the densities of the two populations is just that which D’Ancona observed.