Research on stability and Hopf bifurcation of marine ecosystem dynamics models
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Research on stability and Hopf bifurcation of marine ecosystem dynamics models SHI Honghua1*, HU Long2, FANG Guohong1, WEI Zexun1, SHEN Chengcheng1, 3, LIU Yongzhi4 1 First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China 2 School of Mathematics, Shandong University, Jinan 250100, China 3 College of Environmental Science and Engineering, Ocean University of China, Qingdao 266100, China 4 Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266003, China
Received 20 July 2015; accepted 21 September 2015 ©The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 2016
Abstract
The predictability of marine ecosystem dynamics models is one of the most vital factors to limit their practical applications, of which the stability is the fundamental condition. In order to discuss the stability and Hopf bifurcation of marine ecosystem dynamics models, an approach based on a theorem termed dimension reduction was proposed and further applied in the mass-conservative nutrient-phytoplankton-zooplankton-detritus (NPZD) model in this paper. Results showed that the nonsingular equilibrium point of NPZD model was analytically stable in use of the dimension reduction theorem and the Hopf bifurcation might occur when model parameters changed along the threshold values. The analytical results of the NPZD model were further verified by numerical simulation in this study. It can be concluded that this approach based on the dimension reduction theorem is well applicable to the theoretical analysis of a kind of stability problems and Hopf bifurcation of massconservative systems. Key words: marine ecosystem, mass-conservative, NPZD, stability, Hopf bifurcation Citation: Shi Honghua, Hu Long, Fang Guohong, Wei Zexun, Shen Chengcheng, Liu Yongzhi. 2016. Research on stability and Hopf bifurcation of marine ecosystem dynamics models. Acta Oceanologica Sinica, 35(4): 124–132, doi: 10.1007/s13131-016-0809-x
1 Introduction Human depends on marine ecosystems for their unique and valuable goods and services (Costanza et al., 1997). However, human activities have greatly altered the structure and function of marine ecosystem through direct and indirect means (Jackson et al., 2001; Myers and Worm, 2003; MA, 2005; Lotze et al., 2006). Recent research indicates that no area is unaffected by human activities and there is a large fraction (41%) strongly affected by multiple drivers (Halpern et al., 2008; Halpern et al., 2012). For instance, harmful algae blooms in coastal waters caused by eutrophication in aquatic ecosystem have attracted more and more interest of biologists (Bricker et al., 2003; Fujii et al., 2005; Granéli et al., 2008; Gao et al., 2009). Numerous mathematical models have been developed to investigate the pelagic ecosystem dynamics and nutrient recycling (Franks and Chen, 1996; Gao et al., 1998; Chen et al., 1999; Liu et al., 2007; Vichi et al., 2007). Marine ecosystem dynamics models, which are used to quantitatively describe the dynamic processes of marine ecosyst
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