  The four-year oscillations of the number of spawning sockeye salmon
(Oncorhynchus nerka) that return to their native stream within the Fraser River
basin in Canada are a striking example of population oscillations. The period
of the oscillation corresponds to the dominant generation time of these fish.
Various - not fully convincing - explanations for these oscillations have been
proposed, including stochastic influences, depensatory fishing, or genetic
effects. Here, we show that the oscillations can be explained as a stable
dynamical attractor of the population dynamics, resulting from a strong
resonance near a Neimark Sacker bifurcation. This explains not only the
long-term persistence of these oscillations, but also reproduces correctly the
empirical sequence of salmon abundance within one period of the oscillations.
Furthermore, it explains the observation that these oscillations occur only in
sockeye stocks originating from large oligotrophic lakes, and that they are
usually not observed in salmon species that have a longer generation time.
