Limit Cycles in Predator-Prey Models DARIUS2 M. WRZOSEK Institute of Applied Muthemarics, Deparfmenl of Mathemutics, Corti_vuter Science and Mechanics, University of Warsuw, 00-901 Warsaw, Poland Received IX Fehruaw 1989; revised 24 June I989 ABSTRACT The general model of interaction between one predator and one prey is studied. A unimodal function of rate of growth of the prey and concave down functional response of the predator is assumed. In this work it is shown that for a given natural number n there exist models possessing at least 2n + 1 limit cycles. It is also proved, applying the Hopf bifurcation theorem, that a model exists with a logistic growth rate of the prey and concave down functional response that has at least two limit cycles. 1. INTRODUCTION The existence and number of limit cycles is one of the most delicate problems connected with two-dimensional predator-prey models. Hsu and coworkers [6] considered the model (14 (lb) where V is prey density, P is predator density, and (in the notation from [6]) c = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA m/y, b = y, and d = Do. In that work it is proved that if there exists an asymptotically stable equilibrium point in the positive quadrant it is also globally stable. Cheng shows in [2] that if this equilibrium point is unstable, then it is surrounded by exactly one stable limit cycle. Harrison [4] finds the form of the Lyapunov function for a very general predator-prey model, giving by this means a strong tool for investigation of the global stability of the equilibrium point. Conway and Smoller [3] investigate the so-called Rosenzweig-MacArthur equation with a linear response of the predator to an MATHEMATICAL BIOSCIENCES 98:1-12 (1990) OElsevier Science Publishing Co., Inc., 1990 1 655 Avenue of the Americas, New York, NY 10010 0025-5564/90/$03.50