Using Shannon Entropy and Contagion Index to Interpret Pattern Self-Organization in a Dynamic Vegetation-Sand Model

Feifan Zhang,Lei Yao,Wenjiao Zhou,Qinjing You,Huayong Zhang

doi:10.1109/access.2020.2968242

Feifan Zhang, Lei Yao + Show 3 more

Open Access

https://doi.org/10.1109/access.2020.2968242

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Abstract

Vegetation pattern is one of the most important self-organized patterns in ecological systems. The formation mechanism of vegetation patterns has been attributed to dynamic bifurcations, while from the external perspective, the regularity of patterns could also be influenced by some statistical indicators. Shannon entropy and contagion index are the most commonly used indicators of landscape diversity and connectivity in landscape ecology. These two indicators can explain the self-organization of vegetation patterns. In this research, vegetation patterns are neither randomly generated nor captured from vegetation map. Based on a discrete vegetation-sand model, formation process of vegetation patterns are simulated in different situations of bifurcations. Given different situations of bifurcations (Turing bifurcation, Neimark-Sacker bifurcation and Turing-Neimark-Sacker bifurcation), several formation processes are studied. Along the process, the corresponding Shannon entropy and contagion index of simulated vegetation patterns are calculated based on slightly modified calculation formulas. Comparing different variation curves of Shannon entropy and contagion index, we can see that variation trends of both Shannon entropy and contagion index are closely related to the formation stages of vegetation patterns. The different final values of Shannon entropy and contagion index in different patterns can be used to determine which bifurcation is in dominant when both bifurcations occur.

Highlights

Patterns are non-uniform structures with some regularity in time and space
In this research, based on a discrete vegetation sand model discretized from Zhang et al [26], we will study the variations of Shannon entropy and contagion index in the formation process of vegetation patterns
TURING-NEIMARK-SACKER TYPE SELF-ORGANIZATION In the above sections, we have studied the process of vegetation pattern formation when Turing bifurcation or NeimarkSacker bifurcation occur alone

Summary

INTRODUCTION

Patterns are non-uniform structures with some regularity in time and space. It is the state of ordered distribution caused by the interaction between the parameters in the system in a certain way. By applying the bifurcation dynamics in nonlinear science, the formation of a self-organized pattern of vegetation can be obtained in numerical simulations. The use of dynamic entropy to analyze the self-organizing process is obviously repeated This is the reason why the above research results are not good. We believe that the intrinsic mechanism of the formation of the self-organized pattern of vegetation is the bifurcation dynamics, but at the same time, the geometric structure of the self-organized pattern will be subject to space constraints. In this research, based on a discrete vegetation sand model discretized from Zhang et al [26], we will study the variations of Shannon entropy and contagion index in the formation process of vegetation patterns. The variation trends will be analyzed and compared among different situations

MODEL AND METHOD

SHANNON ENTROPY

CONTAGION INDEX

CALCULATIONS OF SHANNON ENTROPY AND CONTAGION INDEX OF A SIMULATED PATTERN

Findings

DISCUSSION AND CONCLUSION