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

Xin Wei; Jianjun Paul Tian; Jiantao Zhao; Xin Wei; Jianjun Paul Tian; Jiantao Zhao

doi:10.3934/mbe.2024297

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 8: 6783-6806. doi: 10.3934/mbe.2024297

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Research article

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

1.
School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang 150080, China
2.
Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88001, USA

Received: 04 June 2024 Revised: 25 July 2024 Accepted: 01 August 2024 Published: 07 August 2024

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.
- transient fairy circle,
- temporal periodic pattern,
- plant-sulfide feedback,
- time delay,
- Hopf bifurcation
Citation: Xin Wei, Jianjun Paul Tian, Jiantao Zhao. Fairy circles and temporal periodic patterns in the delayed plant-sulfide feedback model[J]. Mathematical Biosciences and Engineering, 2024, 21(8): 6783-6806. doi: 10.3934/mbe.2024297

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Abstract

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.

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