Abstract

We generalize sensitivity analysis to press perturbations for ecological networks described by time-varying ODE systems. Despite that, in a time-invariant framework, sensitivity analysis has been successfully employed to assess the overall effects stemming from sustained changes in species growth rates on the equilibrium values of other, indirectly related species, and its generalization to the time-varying case has remained elusive. Our accomplishments were based on two classical topics of ODEs: (i) the smoothness of solutions with respect to parameters and (ii) the properties of exponential dichotomy and admissibility, which were combined with recent advances in the theory of nonautonomous dynamical systems inspired by the new concept of nonautonomous equilibrium and its local properties. Specifically, we derive the sensitivity matrix as the solution of a nonhomogeneous linear matrix differential equation, which can be explicitly computed by using results from exponential dichotomy and admissibility. In addition, time averaging provides alternative characterizations for the sensitivity matrix in terms of the community one.