Please cite this article in press as: Burns, T.P., et al., Quantifying direct and indirect effects of perturbations using model ecosystems. Ecol. Model. (2014), http://dx.doi.org/10.1016/j.ecolmodel.2013.12.017 ARTICLE IN PRESS G Model ECOMOD-7102; No. of Pages 12 Ecological Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect Ecological Modelling jo ur nal ho me page: www.elsevier.com/locate/ecolmodel Quantifying direct and indirect effects of perturbations using model ecosystems Thomas P. Burns a,∗ , Kenneth A. Rose b , Antoinette L. Brenkert c a Pellissippi State Community College, Knoxville, TN 37933-0990, United States b Department of Oceanography and Coastal Sciences, Louisiana State University, Baton Rouge, LA 70803, United States c 2825 39th Street NW, Washington, DC 20007, United States a r t i c l e i n f o Article history: Available online xxx Keywords: Indirect effect Ecosystem Simulation model Perturbation a b s t r a c t Indirect effects in ecosystems in response to perturbations remain a topic of much discussion. We con- tinue 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. © 2014 Elsevier B.V. All rights reserved. 1. Introduction Indirect effects can play an important role in determining the response of natural ecosystems to perturbations. Theoreti- cal models have shown how indirect effects emerge in simple networks of interacting organisms (MacArthur, 1972; Levine, 1976; Vandermeer, 1980; Higashi and Patten, 1989; Yodzis, 2000; Dambacher and Ramos-Jiliberto, 2007), and empirical studies have demonstrated their existence in nature (e.g., Zaret and Paine, 1973; Archibald, 1975; Carpenter et al., 1985; Fleeger et al., 2003; Montoya et al., 2009; Yang et al., 2010). An easily understood exam- ple of a density-mediated indirect effect (Abrams et al., 1996) is the change in abundance of a carnivore (e.g., game fish) producing a change in the abundance of a species that it does not directly use as ∗ Corresponding author at: 447 East Drive, Oak Ridge, TN 37830, United States. Tel.: +1 865 385 7063. E-mail address: burns50@comcast.net (T.P. Burns). a resource (e.g., phytoplankton) (Hrbacek et al., 1961; Hendrickson et al., 1980; Brönmark and Weisner, 1992; Vanni and Layne, 1997). Indirect effects between species (i.e., populations in a given ecosys- tem) that interact directly, such as the game fish and one of its prey species, can be more subtle and difficult to disentangle (e.g., Werner and McPeek, 1994). Indirect effects are clearly a topic of importance for understanding and managing natural ecosystems (Arkema et al., 2009). If indirect effects are important in determining the dynamics of species in nature, then management models and theories that do not consider them will likely give inaccurate predictions (Abrams et al., 1996). Under what conditions indirect effects are important in deter- mining the dynamics of species and ecosystems is an open question. Using path analysis and structural equation modeling (Johnson et al., 1991; Wootton, 1994; Shipley, 2000, 2009; Clough, 2012), evidence of important indirect effects has been found in forest-floor arthropod food webs by manipulating ant populations (Moya- Lara ˜ no and Wise, 2007) and in herbivorous insect communities from fire-induced changes to oak-scrub plants (Kim and Holt, 0304-3800/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2013.12.017