Open Systems ,~ Information Dynamics Vol. 1 No 1 (1992) 127-147 (~ Nicholas Copernicm University Press Discrete Waves and Phase-Locked Oscillations in the Growth of a Single-Species Population Over a Patchy Environment JIANHONG WU Department of Mathematics and Statistics, York University North York, Ontario, Canada, M3J 1P3 W. KRAWCEWICZ Department of Mathematics, University of Alberta Edmonton, Alberta, Canada, T6G ~G1 (Received February 14, 1992) A system of retarded functional differential equations is proposed as a model of the growth of a single-species population distributed over a ring of identical patches where the dispersion from one patch to others occurs only in nearest neighbors. It is shown that the temporal delay and spatial dispersion can give rise to a special type of discrete wave, the so-called phase-locked oscillation in which a population in one patch oscillates just like others except in different phases. Such oscillations cannot be observed in the absence of dispersion. Our results are based on the theory of symmetric Hopf bifurcations from multiple eigenvalues of differential delay equations recently developed by Geba and the authors. 1 Introduction It is generally recognized that time delays in the growth of a single-species population can arise from a great variety of causes and play a significant role in the long-time behavior of population growth. Oscillations observed in laboratory experiments and in the real world are often attributed to delayed growth rate responses. (See Cushing [9], MacDonald [30], May [31, 32] and references therein.) The subject of the effect of environmental changes on the growth and diffusion of populations in a heterogeneous habitat is also a topic of considerable ecological interest. It has been shown that This research was supported by NSERC-Canada. 127