# Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Allen–Cahn Model

#### 2.2. Cahn–Hilliard Model

#### 2.3. Cahn–Hilliard with Population Demographics Model

## 3. Materials and Methods

#### 3.1. Numerical Simulations

#### 3.2. Spatial Correlation Functions

#### 3.3. Structure Factors

#### 3.4. Density Fluctuation

## 4. Results and Discussion

#### 4.1. Spatial Patterns

#### 4.2. Spatial Correlation Functions

#### 4.3. Structure Factors

#### 4.4. Density Fluctuation

#### 4.5. Growth Law

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AC | Allen–Cahn |

CH | Cahn–Hilliard |

CHPD | Cahn–Hilliard with population demographics |

LS | Lifshitz–Slyozov scaling |

PDE | partial differential equation |

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**Figure 1.**The patterned dynamics of Allen–Cahn model in Equation (1) solved with the simple Euler algorithm starting a random initial condition: (

**a**–

**d**) four typical patterns at $t=0$, $t=500$, $t=5000$, and $t=\mathrm{50,000}$. Numerical simulation was implemented on the discrete $2048\times 2048$ lattices with $\mathsf{\Delta}x=\mathsf{\Delta}y=1.0$ and $\mathsf{\Delta}t=0.0025$. It can be observed that the AC model has a relatively scattered wavelength on the spatial scales. Parameter values for the simulation: $\alpha =0.5$, $\beta =1.0$, and $\epsilon =0.5$.

**Figure 2.**The patterned dynamics of Cahn–Hilliard model in Equation (2) solved with the simple Euler algorithm starting a random initial condition: (

**a**–

**d**) four typical patterns at $t=0$, $t=5000$, $t=$ 50,000, and $t=$ 500,000. Numerical simulation was implemented on the discrete $2048\times 2048$ lattices with $\mathsf{\Delta}x=\mathsf{\Delta}y=1.0$ and $\mathsf{\Delta}t=0.0025$. It can be observed that the CH model has a more concentrated wavelength on the spatial scale than the AC model, which seems to be more regular on the time scales. Parameter values for the simulation: $\alpha =0.5$, $\beta =1.0$, ${\epsilon}_{A}=0.5$, and ${\epsilon}_{B}=1.0$.

**Figure 3.**The patterned dynamics of Cahn–Hilliard with population demographics model solved with the simple Euler algorithm starting a random initial condition: (

**a**–

**d**) four typical patterns at $t=0$, $t=5000$, $t=$ 50,000, and $t=$ 500,000 . Numerical simulation was implemented on the discrete $2048\times 2048$ lattices with $\mathsf{\Delta}x=\mathsf{\Delta}y=1.0$ and $\mathsf{\Delta}t=0.0025$. It can be observed that the pattern of the Cahn–Hilliard with population demographics model is dense at the initial moment, and the dots gradually become spatial regular patterns. The spatial patterns approximate a stable state after the critical time scales. At this time, the coarsening processes is suppressed by the population mortality and birth processes. Parameter values for the simulation: $\alpha =0.5$, $\beta =1.0$, ${\epsilon}_{A}=0.5$, ${\epsilon}_{B}=1.0$, $r=0.0001$, and $K=0.3$.

**Figure 4.**Spatial correlation functions for the self-organized patterns at different times: (

**a**) Allen–Cahn model; (

**b**) Cahn–Hilliard model; and (

**c**) Cahn–Hilliard with population demographics model. Different colors indicate the different times.

**Figure 5.**(

**a**–

**c**) The structure factors $s\left(k\right)$ of the spatial patterns at different times. (

**d**–

**f**) The scaled structure factors $s\left(k\right){k}_{1}^{2}$ versus $k/{k}_{1}$ at different times. Different colors indicate the different times.

**Figure 6.**Spatiotemporal hallmarks of three different dynamics models (Allen–Cahn model ○, Cahn–Hilliard model □, and Cahn–Hilliard with population demographics model Δ). (

**a**) The variable-field fluctuations as a function of the window size length L at $t=\mathrm{500,000}$ associated with $\mathrm{20,000}\times \mathrm{20,000}$ systems. (

**b**) The scaling behavior of the spatial wavelength ${R}_{s}\left(t\right)$ versus increased times.

Model Types | Structure Factors | Density Fluctuation | Growth Law | Saturation |
---|---|---|---|---|

Allen–Cahn | −3.12 | −1.91 | 0.37 | No |

Cahn–Hillard | −3.47 | −2.90 | 0.32 | No |

Cahn–Hillard and logistic term | −5.68 | −2.60 | 0.19 | Yes |

Validity of the indicators | Yes | Yes | Yes | Yes |

Model | Ecosystem | Ecological Processes | Refs. |
---|---|---|---|

AC | Bacteria (Vibrio cholerae) | Phase separation caused by competition | [18] |

Plant (Centaurea maculosa) | Allelopathy and exotic plant invasion | [50] | |

Coral reef | Allelopathy and spatial competition | [51] | |

CH | Mussels | Density-dependent movement behavior | [8,13] |

Ants | Density-dependent movement behavior | [52] | |

Bacteria (Escherichia coli) | Density-dependent chemo-taxis behavior | [53,54] | |

Birds | Resource-dependent movement behavior | [55] | |

Elk (Cervus canadensis) | Socially inform | [56] | |

Zebrafish | Run-and-chase behavior movement | [57] | |

Sperm | Integrated geometry with minima drag | [58] | |

CHPD | Bacteria (Escherichia coli, Bacillus subtilis) | Density-dependent motility and birth-death | [19,20] |

Stones and soil | Freeze–thaw cycles | [59] | |

Insect | General theory | [16] |

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**MDPI and ACS Style**

Zhang, K.; Hu, W.-S.; Liu, Q.-X. Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns. *Mathematics* **2020**, *8*, 112.
https://doi.org/10.3390/math8010112

**AMA Style**

Zhang K, Hu W-S, Liu Q-X. Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns. *Mathematics*. 2020; 8(1):112.
https://doi.org/10.3390/math8010112

**Chicago/Turabian Style**

Zhang, Kang, Wen-Si Hu, and Quan-Xing Liu. 2020. "Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns" *Mathematics* 8, no. 1: 112.
https://doi.org/10.3390/math8010112