Proportional relations between the wave number and amplitude of spiral waves near Neimark-Sacker bifurcations

The spatiotemporal distribution patterns of interacting populations are broadly accepted as a pivotal factor in sustaining species diversity. Spiral waves represent common spatiotemporal patterns observed in ecosystems and biological systems, encompassing both continuoustime and discrete-time systems. The study of the dynamics and regulation of spiral waves in continuous-time systems, often observed in the vicinity of Hopf bifurcations, has been comprehensively examined. However, the dynamical characteristics and rules governing spiral waves near Hopf bifurcations in discrete-time systems, also named Neimark-Sacker bifurcations, are still not fully understood. Here, we investigate spiral waves in a discrete-time predator-pest model caused by a Neimark-Sacker bifurcation. Our results suggest a linear relationship between the amplitude and wave number of spiral waves near the Neimark-Sacker bifurcation. At last, we propose a model that can describe the behaviors of spiral waves in discrete-time systems near Neimark-Sacker bifurcations. Our findings illuminate the process of pattern formation in discrete-time systems, offering potential insights for forecasting and managing pest distribution.