Ebert U, Arrayás M, Temme N, Sommeijer B, Huisman J
Aquatic Microbiology, Institute for Biodiversity and Ecosystem Dynamics, Universiteit van Amsterdam, Nieuwe Achtergracht 127, 1018 WS Amsterdam, The Netherlands.
Bull Math Biol. 2001 Nov;63(6):1095-124. doi: 10.1006/bulm.2001.0261.
We motivate and analyse a reaction-advection-diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton-water-light-system characterized by the same dimensionless parameters will show the same dynamics.
我们构建并分析了一个用于描述浮游植物物种动态的反应-平流-扩散模型。浮游植物的繁殖率由局部光照强度决定。由于水和浮游植物的吸收作用，光照强度随深度而降低。浮游植物在给定深度的水柱中通过湍流扩散进行输运。此外，根据其比重，它可能下沉或上浮。量纲分析可将完整问题简化为一个具有四个无量纲参数的问题，并对其进行全面探究。我们证明，对于固定浮游植物水华不再存在的临界参数区域，可通过具有特定边界条件的简化线性化方程进行分析。该问题可精确映射为一个贝塞尔函数问题，通过数值计算和渐近展开对其进行评估。从无量纲参数到实验室参数的最终转换，得出了一套关于允许浮游植物水华发展条件的完整预测。我们的结果表明，浮游植物水华发展的条件可由一个临界深度、一个补偿深度以及垂直湍流扩散系数的零个、一个或两个临界值来描述。这些可通过实验验证 的预测采用相似律形式：每个以相同无量纲参数为特征的浮游生物-水-光系统将呈现相同的动态。
