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Open AccessArticle

Coupled Effects of Turing and Neimark-Sacker Bifurcations on Vegetation Pattern Self-Organization in a Discrete Vegetation-Sand Model

Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing 102206, China
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Entropy 2017, 19(9), 478; https://doi.org/10.3390/e19090478
Received: 12 July 2017 / Revised: 11 August 2017 / Accepted: 4 September 2017 / Published: 8 September 2017
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
Wind-induced vegetation patterns were proposed a long time ago but only recently a dynamic vegetation-sand relationship has been established. In this research, we transformed the continuous vegetation-sand model into a discrete model. Fixed points and stability analyses were then studied. Bifurcation analyses are done around the fixed point, including Neimark-Sacker and Turing bifurcation. Then we simulated the parameter space for both bifurcations. Based on the bifurcation conditions, simulations are carried out around the bifurcation point. Simulation results showed that Neimark-Sacker bifurcation and Turing bifurcation can induce the self-organization of complex vegetation patterns, among which labyrinth and striped patterns are the key results that can be presented by the continuous model. Under the coupled effects of the two bifurcations, simulation results show that vegetation patterns can also be self-organized, but vegetation type changed. The type of the patterns can be Turing type, Neimark-Sacker type, or some other special type. The difference may depend on the relative intensity of each bifurcation. The calculation of entropy may help understand the variance of pattern types. View Full-Text
Keywords: Turing bifurcation; Neimark-Sacker bifurcation; pattern self-organization; vegetation-sand system; entropy Turing bifurcation; Neimark-Sacker bifurcation; pattern self-organization; vegetation-sand system; entropy
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Zhang, F.; Zhang, H.; Huang, T.; Meng, T.; Ma, S. Coupled Effects of Turing and Neimark-Sacker Bifurcations on Vegetation Pattern Self-Organization in a Discrete Vegetation-Sand Model. Entropy 2017, 19, 478.

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