Pii: S0025-5564(00)00004-3

We studied the dynamics of the Ricker population model under perturbations by the discrete random variable e which follows distribution Pfe ˆ aig ˆ pi; i ˆ 1; . . . ; n; 0 < ai 1, n P 1. Under the perturbations, n‡ 1 blurred orbits appeared in the bifurcation diagram. Each of the n‡ 1 blurred orbits consisted of n sub-orbits. The asymptotes of the n sub-orbits in one of the n‡ 1 blurred orbits were Nt ˆ ai for i ˆ 1; . . . ; n. For other n blurred orbits, the asymptotes of the n sub-orbits were Nt ˆ ai exp ‰r…1ÿ ai†Š ‡ aj; j ˆ 1; 2; . . . ; n, for i ˆ 1; . . . ; n, respectively. The e€ects of variances of the random variable e on the bifurcation diagrams were examined. As the variance value increased, the bifurcation diagram became more blurred. Perturbation e€ects of the approximate continuous uniform random variable and random error were compared. The e€ects of the two perturbations on dynamics of the Ricker model were similar, but with di€erences. Under di€erent perturbations, the attracting equilibrium points and two-cycle periods in the Ricker model were relatively stable. However, some dynamic properties, such as the periodic windows and the n-cycle periods (n > 4), could not be observed even when the variance of a perturbation variable was very small. The process of reversal of the period-doubling, an important feature of the Ricker and other population models observed under constant perturbations, was relatively unstable under random perturbations. Ó 2000 Elsevier Science Inc. All rights reserved.