NORTH-~ On a Model of Competition in Periodic Environments Pingzhou Liu and K. Gopalsamy Department of Mathematics and Statistics The Flinders University of South Australia GPO Box 2100 Adelaide 5001, Australia ABSTRACT Sufficient conditions are obtained for the existence and global attractivity of a positive periodic solution of the system b,(n) x,(~) x,(n+ 1) = 1 + F~a~lcij(n)xj(n )' i= 1,2,...,m; n• N in which {bi(n)}, {ci~(n)} (i, j = 1, 2,..., m), n • N denote positive periodic se- quences with a common period. Also, a method is proposed for the derivation of the discrete system from a continuous time analogue modelling competition in periodic environments with time delays in the stabilizing negative feedback effects. © Elsevier Science Inc., 1997 1. INTRODUCTION Over the past fifteen years, interest in the dynamics of models of competition in temporally inhomogeneous environments has been steadily increasing. Almost all the models of competition considered have been the obvious modifications to time-varying environments of the already familiar models in constant environments, related to the well-known Lotka-Volterra system of ordinary differential equations. We refer to the numerous articles in the literature for several results corresponding to the existence of stability of periodic solutions in periodic environments [1-16]. While continuous-time competition models described by ordinary differential equations correspond APPLIED MATHEMATICS AND COMPUTATION 82:207-238 (1997) © Elsevier Science Inc., 1997 0096-3003/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0096-3003(96)00044-6