Abstract
The coupled set of first-order nonlinear differential equations describing a generalized form of Volterra's problem of two conflicting populations ẋ=C0+C1x+C2y+C3xy+C4x2+C5y2,ẏ=A0+A1x+A2y+A3xy+A4x2+A5y2are solved by an approximate method which gives y(t) for the particular case in which the variables x and y vary periodically, the coefficients Ci and Ai are real, and the peak-to-peak amplitude of x is small compared with the mean value of x. The peak-to-peak amplitude of y, however, is not necessarily small compared with the mean value of y. When these conditions are satisfied, the functional form of y(t) is approximated by Jacobian elliptic functions. The solutions obtained in this analysis are relevant to special cases of the classical problem of predator and prey, and also to certain low-frequency oscillations in partially ionized plasmas that arise from periodic solutions to the neutral and charged-particle continuity equations.
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