Effects of time lags on transient characteristics of a nutrient cycling model

A simple ecosystem with limiting nutrient cycling is modeled by chemostat equations with an integral term describing the continuous time lag involved in the process of nutrient regeneration from organic sediments. The same model has already been proposed in a previous paper, where conditions for boundedness of the solutions and stability of the equilibria were given. This paper is concerned with the relationships between resilience, that is, the speed with which the system returns to a stable equilibrium following a perturbation, and the time lag in the nutrient recycling process. Simple algorithms are given for the numerical calculation of the characteristic return time toward the stable equilibrium following a small perturbation. These methods also allow us to distinguish the case of monotone convergence from that of oscillatory convergence toward equilibrium. The numerical results obtained show that the presence of the time lag causes both qualitative and quantitative modifications in the dependence of equilibrium resilience on some relevant ecological parameters, such as the input nutrient concentration and the recycling extent. Analytical results for "quasi-closed" ecosystems are given that show that such stable systems are characterized by a very low resilience.