Mathematical analysis on coexistence conditions of phytoplankton and bacteria systems with nutrient recycling

We analyzed a condition of dynamical coexistence of phytoplankton and bacteria in the model systems with nutrient recycling from the viewpoint of permanence. In this model, nutrients were classified into dissolved and particulate matter, and bacteria use not only particulate nutrient but also dissolved nutrient. We led the sufficient conditions of permanence in the model system. On the basis of these conditions, we studied how they can achieve the coexistence estimating indirect effects between phytoplankton and bacteria through nutrients, and argued the effects of nutrient recycling through bacterial decomposition on the coexistence. From these analysis, we obtained that coexistence conditions can be explained by invasion conditions, and coexistence can be achieved even if bacteria cannot always invade the steady state consisting of only two types of nutrient. The estimation of indirect effects indicated the mutualistic interactions between phytoplankton and bacteria under the conditions of permanence. This suggests that mutualistic situation will be occurred by nutrient recycling after sufficiently long time, even if phytoplankton and bacteria compete with each other through common resource initially.