Subharmonic solutions of periodically forced predator-prey systems

Abstract

IN THIS PAPER we consider a class of planar ordinary differential equations consisting of a basic predator-prey system with an autonomous perturbation that may represent a damping or “frictional” force and a time-periodic perturbation. The problem we study is that of the existence of subharmonic solutions, that is, periodic solutions whose period is an integral multiple of the forcing period. It is not our intention to describe any specific ecosystem, but rather to give some rather general mathematical results and discuss the numerical computation of these subharmonics. We mention, however, that predator-prey cycles have been observed in nature for a long time, and their mechanism remains a major problem in mathematical biology. Some classic early studies were those of Volterra [l], Lotka [2] and Kolmogoroff [3], and the examples in Section 4 of this paper are perturbations of Volterra-Lotka systems. Considerable work has been done with various autonomous predator-prey models (see [4, Chapter 121 for a very readable introduction), but much less with time-dependent systems. From the point of view of applications, however, it seems reasonable to introduce periodic perturbations, and indeed it is difficult to imagine any high-level ecosystem in which the reproduction rates of at least some species are not affected by some cyclic factor (e.g. yearly, daily, lunar). It is hoped that the results given here may contribute to the understanding of such systems. For discussions of predator-prey cycles from the biological point of view, see [5-71. The main tool here is a variant of the Liapunov-Schmidt method as given by Chow and Hale [8]. The Liapunov-Schmidt method first seems to have been used to prove the existence of subharmonics for some classes of ordinary differential equations by Hale and Taboas [9]. Since then, a fairly extensive theory has been developed for single second-order equations. (see [lo] and references therein.) Subharmonics for a certain perturbed Volterra-Lotka system were examined from a different point of view in [ 111. We will now make a few comments on our notation. Whenever a function is introduced on the right-hand side of a differential equation, it is assumed to be smooth. “Smooth” here means C”. It will be evident that a hypothesis of C” in what follows can usually be replaced by a hypothesis of C” for some finite n, but we wanted to avoid the distraction of counting derivatives. Ordinary derivatives are denoted in the usual way by primes, dots, or d”, partial derivatives by subscripted variable names or a’, and derivative matrices by Ds. The 2 x 2 identity matrix is denoted by I. A “wedge” is a section of a disk lying between two smooth curves through the origin whose tangents at the origin are different.