Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns

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.

Fourier-Spectral Method for the Phase-Field Equations

“…Such models are rewritten to sum of linear operator part with high-order terms and nonlinear operator part, and then Fourier-spectral method is applied in order to employ exponential integrator to this semilinear ordinary differential equations. Zhang and Liu [16] used several AC or CH type equations to represent the spatial patterns in ecological and biological system. Shen and Yang [17] presented numerical approximations of the AC and CH equations for semi-implicit or implicit schemes which are unconditionally energy stable, with stability analysis and error estimates based on spectral-Galerkin method.…”