Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick

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• 作者：Stephen Ippolito ; Vincent Naudot ; Erik G. Noonburg
• 关键词：Coral reefs ; Alternative stable states ; Dirac delta distribution ; Dynamical systems ; Competitive dynamics
• 刊名：Bulletin of Mathematical Biology
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：78
• 期：3
• 页码：413-435
• 全文大小：2,426 KB
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• 作者单位：Stephen Ippolito (1)
  Vincent Naudot (1)
  Erik G. Noonburg (2)
  
  1. Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL, 33431, USA
  2. Department of Biological Sciences, Florida Atlantic University, 3200 College Ave., Davie, FL, 33314, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Biology
  
• 出版者：Springer New York
• ISSN：1522-9602

文摘

We consider a computer simulation, which was found to be faithful to time series data for Caribbean coral reefs, and an analytical model to help understand the dynamics of the simulation. The analytical model is a system of ordinary differential equations (ODE), and the authors claim this model demonstrates the existence of alternative stable states. The existence of an alternative stable state should consider a sudden shift in coral and macroalgae populations, while the grazing rate remains constant. The results of such shifts, however, are often confounded by changes in grazing rate. Although the ODE suggest alternative stable states, the ODE need modification to explicitly account for shifts or discrete events such as hurricanes. The goal of this paper will be to study the simulation dynamics through a simplified analytical representation. We proceed by modifying the original analytical model through incorporating discrete changes into the ODE. We then analyze the resulting dynamics and their bifurcations with respect to changes in grazing rate and hurricane frequency. In particular, a “kick” enabling the ODE to consider impulse events is added. Beyond adding a “kick” we employ the grazing function that is suggested by the simulation. The extended model was fit to the simulation data to support its use and predicts the existence cycles depending nonlinearly on grazing rates and hurricane frequency. These cycles may bring new insights into consideration for reef health, restoration and dynamics.