Sensitivity and stability of flow networks

Abstract A simple relationship between the community matrix and the corresponding sensitivity matrix, the matrix whose (i,j) element represents the sensitivity in storage of compartment i to an addition of inflow to compartment j, is derived through the consideration of an ecosystem under “press” perturbation experiments. It is shown that a superposition rule holds for the change in steady-state storage of a compartment due to addition of inflows to more than one compartment. It is also proved that the sensitivity of compartment i to a parameter change can always be decomposed into the product of the direct dependence on the parameter of the net flows into the compartments that have this dependency and the sensitivities of compartment i to the addition of inflows to these compartments. For donor-dependent systems, it is found that the inflow-sensitivities of any compartment are all non-negative (i.e., an addition of inflow to any compartment always leads to an increase ind storage of every compartment), and that the compartment having an addition of inflow receives, among all the compartments in the system, the greatest impact in storage from that addition of inflow. The notion of resistance is proposed to be defined as the inverse of the sensitivities to press perturbations. For donor-dependent systems, it is proved that this notion of resistance, the stability to press perturbation, consists, with that of resilience, the stability to “pulse” perturbation.