As três leis de Volterra, cem anos depois

In 1926, Vito Volterra published a brief communication in Nature in which he formulated three laws for the dynamics of two interacting species. The paper presented the results with few equations and almost no derivations. In this review article, we provide a complete Portuguese translation of that communication, prepared directly from the facsimile and kept separate from the commentary, and reconstruct the mathematics underlying the three laws. We show that they rely on distinct structures: periodicity follows from the first integral and the geometry of its level curves; time averages follow from balances of logarithmic per-capita rates; and the response to removal follows from changes in per-capita rates, whereas the effect of protection depends on how the intervention modifies the interactions. Linearization further yields the period, phase lag, and amplitude ratio for small oscillations. This decomposition makes it possible to examine separately what remains when the classical model is modified. In generalized Lotka–Volterra systems, for example, the balance of time averages may survive even without periodicity, whereas regulation, saturation, forcing, memory, finite populations, and space alter different parts of the original structure. The 1926 communication is therefore used not only as a historical document, but also as a starting point for identifying the assumptions that support the three laws and the limits of their extension to later models.