We seek to understand, in mathematical terms, the causes of stability in marine phytoplankton biomass. The stability of a simple, mixed-layer-phytoplankton-nutrient model is analysed. Primary production is modelled as a non-linear function of nutrient concentration and light. The steady state of the model system is demonstrated to be stable with a linear relation between steady state biomass and nutrients. The causes of stability are shown to be shading and nutrient limitation. When only light limitation and shading are taken into account, the steady state is a sink node. However, when nutrient limitation is taken into account, without shading, the steady state can be either a sink node or a spiral sink. The transition from a sink node to a spiral sink occurs when normalized mixed layer production becomes larger than the equivalent influx rate of nutrients into the mixed layer, demonstrating that nutrient limitation of production is a necessary, but not a sufficient condition for oscillatory solutions. In both cases, the characteristic return times are derived explicitly. The effect of shading is found to cause the depression of the steady state towards lower biomass than would otherwise be attainable. The influence of mixed-layer depth variation on stability is also analysed.

中文翻译：

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营养性浮游植物海洋生态系统模型的稳定性和复原力

我们试图以数学的方式理解海洋浮游植物生物量稳定的原因。分析了一个简单的混合层浮游植物营养模型的稳定性。初级生产被建模为养分浓度和光的非线性函数。模型系统的稳态被证明是稳定的，并且稳态生物量和养分之间具有线性关系。稳定的原因显示为遮荫和营养限制。当仅考虑光限制和阴影时，稳态为宿节点。但是，当考虑到养分限制而没有阴影时，稳态可以是汇点或螺旋汇。当标准化的混合层产量变得大于养分向混合层中的等效流入量时，就会发生从汇点到螺旋汇的过渡，这表明养分限制生产是振荡解决方案的必要条件，但不是充分条件。在这两种情况下，特征返回时间都是明确得出的。发现遮蔽的作用导致稳态降低，使生物量低于原本可达到的生物量。还分析了混合层深度变化对稳定性的影响。发现遮蔽的作用导致稳态降低，使生物量低于原本可达到的生物量。还分析了混合层深度变化对稳定性的影响。发现遮蔽的作用导致稳态降低，使生物量低于原本可达到的生物量。还分析了混合层深度变化对稳定性的影响。