Modeling on an ecological food chain with recycling

We propose two nutrient-phytoplankton models with instantaneous and time delayed recycling, investigate the dynamics and examine the responses to model complexities. Instead of the familiar specific uptake rate and growth rate functions, we assume only that the nutrient uptake and phytoplankton growth rate functions are positive, increasing and bounded above. We use geometrical and analytical methods to find conditions for the existence of none, one, or at most two positive steady states and analyze the stability properties of each of these equilibria. With the variation of parameters, the system may lose its stability and bifurcation may occur. We study the occurrence of Hopf bifurcation and the possibility of stability switching. Numerical simulations illustrate the analytical results and provide further insight into the dynamics of the models, biological interpretations are given.