加性噪声和周期力作用下空间扩展浮游植物-浮游动物系统的共振和锁频现象
Resonance and frequency-locking phenomena in spatially extended phytoplankton-zooplankton system with additive noise and periodic forces
摘要：
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，如外部周期性的力量，噪音，和扩散过程。空间扩展的浮游植物-浮游动物系统来自Scheffer的原始研究[M Scheffer，Fish and nutritious interplay determines algae biomass：a minimum model，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。研究结果还表明，噪声和周期外力对Scheffer模型有着积极的作用：首先，当噪声强度在一定范围内时，噪声可以增强浮游植物种群密度的振荡，在空间上形成大的集群;第二，外部周期力可以诱导4：1和1：1频率锁定和空间均匀振荡现象出现。最后，在空间噪声和外加周期力的共同作用下，观察到了系统的共振模式，并且发现随着噪声强度的增加，系统的4：1锁频转变为1：1锁频。除了在Turing不稳定性域之外，我们还利用Maple软件进行了数值计算，进一步分析了Turing线性稳定性。值得注意的是，当噪声项存在时，系统中的振荡被增强。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之间的相互作用，而不是确定性因素。这些结果也可能有助于我们理解海洋浮游生物水华不可否认的随机波动所产生的影响。
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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