Abstract
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.
Highlights
The reaction–diffusion systems have been posed to address the spatially extended interactions among species or chemical substances since 1937 [1], when the traveling waves behaviors of the reaction–diffusion equations were discovered and studied
It is notable that CH model describes the mass-conservation ecological processes that remarkably differ from the Allen–Cahn model despite both models reveal a coarsening behavior
partial differential equations (PDEs) models have been widely used to describe the spatial patterns in many ecological and biological systems, few studies focus on inferring the linkage between these visual patterned characters and their potential mechanisms [41,42,43]
Summary
The reaction–diffusion systems have been posed to address the spatially extended interactions among species or chemical substances since 1937 [1], when the traveling waves behaviors of the reaction–diffusion equations were discovered and studied. The spatial systems are widely used to explore the spatiotemporal behaviors going beyond the initial frameworks such as existence and stability of spatial solutions in biology, geology, physics, and ecology [2,3,4,5,6,7,8,9]. The solutions to reaction–diffusion equations display diverse behaviors including the formation of traveling waves [10], wave-like phenomena [11], spatial self-organized patterns [12], and coarsening behaviors [13].
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