Quantifying direct and indirect effects of perturbations using model ecosystems

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Indirect effects in ecosystems in response to perturbations remain a topic of much discussion. We continue on the theoretical path set by Nakajima and Higashi (1995, Indirect effects in ecological interaction networks (II): the conjugate variable approach. Mathematical Biosciences 130, 129–150) and extend their approach for quantifying indirect effects of press perturbations under steady-state conditions to pulse perturbations in steady state and to press and pulse perturbations under time-varying conditions. We illustrate with a commonly used lake ecosystem model how to estimate total, direct, and indirect effects on one species’ biomass (e.g., game fish) from perturbations to a second species (e.g., forage fish). We use the daily output of species biomasses from multi-year model simulations to test whether the computed total, direct, and indirect effects are equal for pulse and press perturbations of equal magnitude, and whether the types (pulse and press) and timing of perturbations alter the relative importance of indirect effects. Our results were consistent with the original theory of Nakajima and Higashi; under steady-state conditions, press and pulse perturbations to the forage fish produced similar effects on the game fish. Under these same conditions, indirect effects on the game fish were more than 3.5 times larger than the direct effects. Under time-varying conditions, all classes of pulse effects, but not press, depended on the day the perturbation was imposed. Indirect effects of pulse and press perturbations under time-varying conditions were always negative (offsetting) and of similar magnitude as the direct effects. When forage fish biomass was growing rapidly, the indirect effect on game fish biomass of a pulse perturbation was relatively smaller compared to the direct effect, whereas the indirect and direct effects were similar at other times. We consider the general implications of these results for the analysis of natural ecosystems.