Bulletin ofMathematicalBiology Vol.45, No. 6, pp. 991-1004, 1983. Printedin Great Britain 0092-8240/8353.00+ 0.00 Pergamon PressLtd. © 1983 Society for Mathematical Biology THE TRADE-OFF BETWEEN MUTUAL INTERFERENCE AND TIME LAGS IN PREDATOR-PREY SYSTEMS? • H.I. FREEDMAN and V. SREE HARI RAO~ Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 We present a Gause predator-prey model incorporating mutual interference among predators, a density-dependent predator death rate and a time lag due to gestation. It is well known that mutual interference is stabilizing, whereas time delays are destabil- izing. We show that in combining the two, a long time-lag usually, but not always, de- stabilizes the system. We also show that increasing delays can cause a bifurcation into periodic solutions. 1. Introduction. There is much literature on predator-prey systems modeled as either a system of autonomous ordinary differential equations, a system of difference equations, a system of integrodifferential equations or a system of differential-difference equations [see Freedman (1980) and the references therein]. By including certain parameters or varying certain assumptions, these models can be made to simulate various types of bio- logical behavior. We note that some models incorporate mutual interference among the predators and/or density-dependent death rates. Others have incorporated time delays (both finite and infinite) in one or more of their functions. To the best of our knowledge none of the work to date has incorporated all of these considerations and analyzed the resulting stability. Predator-prey models incorporating mutual interference were first pro- posed in Hassell (1971) and Rogers and Hassell (1974). A model incor- porating density-dependent death rates was considered by Levin (1977). Equilibrium stability in a predator-prey model incorporating both was analyzed in Freedman (1979). In general, it is found that either or both of these are stabilizing and that intersecting isoclines to the left of a maximum no longer necessarily mean instability of the equilibrium (see Rosenzweig and MacArthur, 1963). Time delays of one type or another have been incorporated by many authors (see, e.g., Wangersky and Cunningham, 1957; Reddingius, 1963; ~Researeh for this paper was partly supported by the Natural Science and Engineering Council of Canada, grant No. NSERC A4823, and by a grant from the University Grants Commission, New Delhi, India, grant No. F. 23-1174/79 (S.R. II). ~: On leave from O smania University, Hyderabad-5 00007, India. 991