Stability and resilience in a nutrient-phytoplankton marine ecosystem model

Abstract

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.