Stability and dynamic transition of vegetation model for flat arid terrains

Lan Jia; Liang Li

doi:10.3934/dcdsb.2021189

Article Contents

2022, Volume 27, Issue 6: 3375-3398. Doi: 10.3934/dcdsb.2021189

This issue Previous Article Matrix measures, stability and contraction theory for dynamical systems on time scales Next Article Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Stability and dynamic transition of vegetation model for flat arid terrains

Lan Jia and
Liang Li^,

College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

^* Corresponding author: Liang Li
^* Corresponding author: Liang Li

Received: October 2020

Revised: June 2021

Early access: July 2021

Published: June 2022

This research was supported by the NSFC, Grant No.11901408 and 11771306

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Abstract

In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an $ S^{1} $ attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.

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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B09, 35K40.

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Figure 1. the topological structure of phase portrait of continuous transition as control parameter $ \lambda >\Lambda_{c} $.

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Figure 2. the topological structure of phase portrait of jump transition as control parameter $ \lambda<\Lambda_{c}. $

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Figure 3. the topological structure of phase portrait of jump transition as control parameter λ < Λ_c

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Figure 4. the topological structure of phase portrait of jump transition as control parameter λ > Λ_c

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Figure 5. 8 nondegenerate singular points

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Figure 6.

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Figure 7. Plot of the numerical solution $ b $ at $ T = 50 $ (left) and plot of bifurcated solution $ \widetilde{b} $ (right) of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 1.00057. $

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Figure 8. Plot of the numerical solution $ b $ and $ w $ at $ T = 600 $ of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 1.0004957. $

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Figure 9. Plot of the numerical solution $ b $(left) at $ T = 600 $ and plot of bifurcated solution $ \widetilde{b} $ (right) of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 16.704876. $

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