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

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.

Keywords

• Contagion index
• Neimark-Sacker bifurcation
• Shannon entropy
• Turing bifurcation