On a nonlocal reaction–diffusion–advection equation modelling phytoplankton dynamics
On a nonlocal reaction–diffusion–advection equation modelling phytoplankton dynamics
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DOI：
10.1088/0951-7715/24/1/016
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发表时间：
2010
期刊：
影响因子：
1.7
通讯作者：
Yihong Du;L. Mei
中科院分区：
文献类型：
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作者：
Yihong Du;L. Mei
我们研究了一个反应-扩散-平流方程，它模拟了单一浮游植物物种在富营养化垂直水柱中的动力学。首先，我们推广了Du和Hsu(2010 SIAM J.Math.分析42 1305-33)，以表明即使在扩散和下沉速率可变的情况下，该模型的全球动力学完全由其唯一的稳态解决定。这意味着Ryabov等人(2010年J.Theor)通过数值模拟观察到的双稳态行为。比奥尔。263 120-33)，因为浮游植物动力学只有在假设模型中的营养物质有限时才会发生。其次，我们分别考察了小扩散、大扩散和深水柱的正定态解的渐近分布。结果表明，对于小扩散，浮游植物种群集中在水柱底部，而对于大扩散，浮游植物种群在水柱中趋于均匀分布，在其它因素相同的情况下，在背景为正浑浊的水体中，大扩散情况下的总生物量大于小扩散情况下的总生物量，在背景为零(或可忽略)的水柱中，总生物量趋于相同的极限；当水柱深度达到无穷大时，种群分布接近石井和高木(1982 J.Math)的结果。比奥尔。161-24)，具有无限水深，并且在一定的有限水位下达到唯一的最大值。我们还完整地回答了徐和楼(2010 SIAM J.Appl.)中的一个未决问题。数学课。70 245-54)关于深水柱的临界死亡率的行为，它在确定是否存在临界水深方面起着关键作用。
We investigate a reaction–diffusion–advection equation that models the dynamics of a single phytoplankton species in a eutrophic vertical water column. First, we extend the results of Du and Hsu (2010 SIAM J. Math. Anal.42 1305–33) to show that even with variable diffusion and sinking rates, the global dynamics of the model is completely determined by its unique steady-state solution. This implies that the bistable behaviour observed through numerical simulation in Ryabov et al (2010 J. Theor. Biol. 263 120–33) for the phytoplankton dynamics can only occur when one assumes limitation of nutrients in the model. Second, we examine the asymptotic profiles of the positive steady-state solution for small diffusion, large diffusion and deep water column, respectively. Our results reveal that for small diffusion, the phytoplankton population concentrates at the bottom of the water column, while for large diffusion, the population tends to distribute evenly in the water column, and when all the other factors are the same, in a water column with positive background turbidity, the total biomass is bigger in the large diffusion case than in the small diffusion case, and in a water column with zero (or negligible) background turbidity, the total biomass tends to the same limit in both cases; when the water column depth goes to infinity, the population distribution approaches that obtained in Ishii and Takagi (1982 J. Math. Biol. 16 1–24) with infinite water depth, and it reaches a unique maximum at a certain finite water level. We also give a complete answer to a question left open in Hsu and Lou (2010 SIAM J. Appl. Math. 70 245–54) regarding the behaviour of the critical death rate for deep water column, which plays a key role in determining whether a critical water depth exists.