Mathematical analysis of coral reef models

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:416

Article

Li, Xiong¹  Wang, Hao²  Zhang, Zheng¹  Hastings, Alan³ 

[1] Beijing Normal Univ, Minist Educ, Lab Math & Complex Syst, Sch Math Sci, Beijing 100875, Peoples R China

[2] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada

[3] Univ Calif Davis, Dept Environm Sci & Policy, Davis, CA 95616 USA

关键词: Coral reef;    Delay differential equation;    Stability;    Hopf bifurcation;   

DOI  :  10.1016/j.jmaa.2014.02.053

来源: Elsevier

【 摘 要 】

It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free