A simple food chain with a growth inhibiting nutrient

We study the dynamics of a simple food chain consisting of a nutrient, prey and predator, where the nutrient is growth inhibiting at high concentrations. The Michaelis-Menten-Monod form for the nutrient uptake rate is generalized to a nonmonotone uptake rate. It is shown that the positive equilibrium is a global attractor for low initial concentrations of the nutrient, i.e., when there is no inhibition effect. However, the behavior of the system can be initial condition dependent at high initial concentrations of the nutrient; persistence may not occur. Several thresholds are defined in terms of the model parameters which determine the global dynamics of the system.