Quite independently, two basic advances were made in the theory of animal populations by A.J. Lotka in 1925 and by Vito Volterra in 1926 and again in 1931. The resulting dynamical problem has implications for a wide range of cases, involving as it does, the possibilities of the extinction of one species, or the evolution to an equilibrated coexistence, or to a continuing oscillation. The problem was brought to Volterra by Ugo d’Ancona, who had found clear evidence for continued oscillation in the type of fish catches in the upper Adriatic. Volterra formulated and solved a system of two non-linear differential equations, in which the proportionate growth rate in each depended only on the level of the other. His solution is stable and positive, but is inappropriate for applied analysis, since it is structurally unstable. It is inappropriate in the sense that a mathematician may assume, as he does, a parameter to be exactly zero, but it is impermissible in an applied or empirical analysis like that of animal populations. In 1931 the Russian mathematician Kolmogoroff gave an elegant generalization which was both dynamically and structurally stable, yielding the three types of solution above, including a demonstration of a stable equilibrium motion, that is, a limit cycle. Various elaborations, along with qualitative analysis in phase space, are given in Hirsch and Smale (1974).
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