Frequency modes of unstable spiral waves in two-dimensional Rosenzweig–MacArthur ecological networks

Abstract

The existence of two frequency regimes in a two-dimensional (2D) Rosenzweig–MacArthur ecological network is debated. The semi-discrete approximation differentiates the two regimes, each described by a 2D complex Ginzburg–Landau equation. Using the standard theory of the linear stability analysis, a generalized expression for the modulational instability growth rate is derived for each frequency mode. The parametric study of the growth rate of modulational instability reveals its sensitivity to the changes in the recruitment rate of the resources. Moreover, direct numerical simulations are carried out to confirm our analytical results. Over the prolonged evolution of the perturbed plane wave solution, the high-frequency mode entertains spiral wave patterns. In contrast, the appearance of target waves manifests the low-frequency regime. In that context, we further explore the impact of the recruitment rate of resources and give the qualitative meaning of the obtained dynamical behaviors and their ecological implications. This work may additionally provide more insight into the mechanism leading to spiral and target waves in environmental systems.