摘要翻译：
在本文中，我们提出了一个空间版本的浮游植物-浮游动物模型，包括一些重要的因素，如外部周期力，噪声和扩散过程。空间扩展的浮游植物-浮游动物系统来源于Scheffer的原始研究[M Scheffer,鱼和营养物的相互作用决定藻类生物量：一个最小模型，Oikos\textbf{62}（1991）271-282]。我们的结果表明，空间扩展系统表现出共振模式和频率锁定现象。该系统还表明，噪声和外部周期力在Scheffer模型中起着构造作用：首先，当噪声强度在一定区间内时，噪声能增强浮游植物物种密度的振荡，并在空间中形成较大的簇。第二，外部周期力能引起4：1和1：1频率锁定和空间均匀振荡现象。最后，当空间噪声和外部周期力同时开启时，观察到系统中的共振模式。此外，我们发现当噪声强度增加时，4:1锁频转变为1：1锁频。除了阐明我们在图灵不稳定性范围之外的结果外，我们还借助Maple软件的数值计算进一步分析了图灵线性稳定性。显著的是，当噪声项出现时，系统中的振荡会增强。这些结果表明，海洋浮游生物水华可能部分是由于随机因素和外力的相互作用而不是确定性因素所致。这些结果也有助于我们理解海洋浮游生物水华的随机波动所产生的不可否认的影响。
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英文标题：
《Resonance and frequency-locking phenomena in spatially extended
  phytoplankton-zooplankton system with additive noise and periodic forces》
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作者：
Quan-Xing Liu, Bai-Lian Li and Zhen Jin
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最新提交年份：
2007
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分类信息：

一级分类：Quantitative Biology        数量生物学
二级分类：Populations and Evolution        种群与进化
分类描述：Population dynamics, spatio-temporal and epidemiological models, dynamic speciation, co-evolution, biodiversity, foodwebs, aging; molecular evolution and phylogeny; directed evolution; origin of life
种群动力学；时空和流行病学模型；动态物种形成；协同进化；生物多样性；食物网；老龄化；分子进化和系统发育；定向进化；生命起源
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一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics        物理学
二级分类：Pattern Formation and Solitons        图形形成与孤子
分类描述：Pattern formation, coherent structures, solitons
图案形成，相干结构，孤子
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一级分类：Quantitative Biology        数量生物学
二级分类：Other Quantitative Biology        其他定量生物学
分类描述：Work in quantitative biology that does not fit into the other q-bio classifications
不适合其他q-bio分类的定量生物学工作
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英文摘要：
  In this paper, we present a spatial version of phytoplankton-zooplankton model that includes some important factors such as external periodic forces, noise, and diffusion processes. The spatially extended phytoplankton-zooplankton system is from the original study by Scheffer [M Scheffer, Fish and nutrients interplay determines algal biomass: a minimal model, Oikos \textbf{62} (1991) 271-282]. Our results show that the spatially extended system exhibit a resonant patterns and frequency-locking phenomena. The system also shows that the noise and the external periodic forces play a constructive role in the Scheffer's model: first, the noise can enhance the oscillation of phytoplankton species' density and format a large clusters in the space when the noise intensity is within certain interval. Second, the external periodic forces can induce 4:1 and 1:1 frequency-locking and spatially homogeneous oscillation phenomena to appear. Finally, the resonant patterns are observed in the system when the spatial noises and external periodic forces are both turned on. Moreover, we found that the 4:1 frequency-locking transform into 1:1 frequency-locking when the noise intensity increased. In addition to elucidating our results outside the domain of Turing instability, we provide further analysis of Turing linear stability with the help of the numerical calculation by using the Maple software. Significantly, oscillations are enhanced in the system when the noise term presents. These results indicate that the oceanic plankton bloom may partly due to interplay between the stochastic factors and external forces instead of deterministic factors. These results also may help us to understand the effects arising from undeniable subject to random fluctuations in oceanic plankton bloom.
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PDF链接：
https://arxiv.org/pdf/0705.3724