Fairy circles and temporal periodic patterns in the delayed plant-sulfide feedback model

Incorporating the self-regulatory mechanism with time delay to a plant-sulfide feedback system for intertidal salt marshes, we proposed and studied a functional reaction-diffusion model. We analyzed the stability of the positive steady state of the system, and derived the sufficient conditions for the occurrence of Hopf bifurcations. By deriving the normal form on the center manifold, we obtained the formulas determining the properties of the Hopf bifurcations. Our analysis showed that there is a critical value of time delay. When the time delay is greater than the critical value, the system will show asymptotical temporal periodic patterns while the system will display asymptotical spatial homogeneous patterns when the time delay is smaller than the critical value. Our numerical study showed that there are transient fairy circles for any time delay while there are different types of fairy circles and rings in the system. Our results enhance the concept that transient fairy circle patterns in intertidal salt marshes can infer the underlying ecological mechanisms and provide a measure of ecological resilience when the self-regulatory mechanism with time delay is considered.