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Open AccessArticle

Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns

by Kang Zhang 1, Wen-Si Hu 2 and Quan-Xing Liu 1,2,3,*
1
School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
2
State Key Laboratory of Estuarine and Coastal Research, School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
3
Shanghai Key Lab for Urban Ecological Processes and Eco-Restoration & Center for Global Change and Ecological Forecasting, East China Normal University, Shanghai 200241, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 112; https://doi.org/10.3390/math8010112 (registering DOI)
Received: 18 November 2019 / Revised: 2 January 2020 / Accepted: 7 January 2020 / Published: 10 January 2020
(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)
Although the diversity of spatial patterns has gained extensive attention on ecosystems, it is still a challenge to discern the underlying ecological processes and mechanisms. Dynamical system models, such partial differential equations (PDEs), are some of the most widely used frameworks to unravel the spatial pattern formation, and to explore the potential ecological processes and mechanisms. Here, comparing the similarity of patterned dynamics among Allen–Cahn (AC) model, Cahn–Hilliard (CH) model, and Cahn–Hilliard with population demographics (CHPD) model, we show that integrated spatiotemporal behaviors of the structure factors, the density-fluctuation scaling, the Lifshitz–Slyozov (LS) scaling, and the saturation status are useful indicators to infer the underlying ecological processes, even though they display the indistinguishable spatial patterns. First, there is a remarkable peak of structure factors of the CH model and CHPD model, but absent in AC model. Second, both CH and CHPD models reveal a hyperuniform behavior with scaling of −2.90 and −2.60, respectively, but AC model displays a random distribution with scaling of −1.91. Third, both AC and CH display uniform LS behaviors with slightly different scaling of 0.37 and 0.32, respectively, but CHPD model has scaling of 0.19 at short-time scales and saturation at long-time scales. In sum, we provide insights into the dynamical indicators/behaviors of spatial patterns, obtained from pure spatial data and spatiotemporal related data, and a potential application to infer ecological processes. View Full-Text
Keywords: Allen–Cahn model; Cahn–Hilliard model; spatial patterns; spatial fluctuation; dynamic behaviors Allen–Cahn model; Cahn–Hilliard model; spatial patterns; spatial fluctuation; dynamic behaviors
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Zhang, K.; Hu, W.-S.; Liu, Q.-X. Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns. Mathematics 2020, 8, 112.

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