Regime shift in a phytoplankton-phosphorus model with vertical structure and seasonality

Abstract

Many ecological systems are influenced by positive feedbacks between organisms and abiotic environments, which generates multiple stable equilibria of a mathematical model with a hysteresis structure. In addition, discontinuous shifts of system at equilibrium is predicted, which is often called regime shift in ecosystem sciences. This hysteresis structure is unfavorable from environmental management point of view, because the reconstruction of original equilibriumstate requestsmuch lower levels of external forcing. Mathematicalmodels proposed in previous works are simple andmathematically tractable ([7], [2]). However, it is difficult to extrapolate from such simple models the occurrence likelihood of regime shift in natural environments since temporally dynamic features in ecology and physico-chemical environments, and spatial dimension are less explored in those models. In this study, we construct a realistic but mathematically tractable model of interaction between phytoplankton and phosphorus, which incorporates (1) 1-dimensional vertical structure of lake ecosystem and (2) seasonal periodic cycle of watermixing. We aim to understand the impact of changes in seasonality in various types of lakes on the occurrence of multiple attractors (periodic solution) and hysteresis structure.