Ecological Monographs, 85(4), 2015, pp. 625–641 Ó 2015 by the Ecological Society of America The many potential indirect interactions between predators that share competing prey PETER A. ABRAMS 1 AND MICHAEL H. CORTEZ 2,3 1 Department of Ecology and Evolutionary Biology, 25 Harbord Street, University of Toronto, Ontario M5S 3G5 Canada 2 Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322 USA Abstract. Using a Lotka-Volterra model, we explore how the indirect interactions between two predators are altered by interspecific competition between two shared prey. We identify when different indirect interactions arise between the predators, classifying interactions by predator responses to (1) slightly increased mortality in the other predator, (2) a slightly decreased population of the other predator, or (3) removal of the other predator. When interspecific prey competition is low, all methods predict negative indirect effects between predators, i.e., competitive interactions. Strong and/or highly asymmetric interspe- cific prey competition often produces at least one positive indirect effect between predators, i.e., mutualism or contramensalism. However, the three methods often disagree about the strength of and signs characterizing the indirect effects between predators, including cases where all three methods predict a qualitatively different interaction. These differences arise for a variety of reasons, including hydra effects (where a predator increases in abundance with increased mortality) and extinction of prey species following the removal of one predator. We also show that cyclic dynamics do not arise in our model when there is a single predator, but under strong interspecific prey competition, the indirect interactions between two predators can drive cyclic community dynamics. Similar phenomena are likely to occur in other food webs, and understanding them will be required to predict the impact of environmental change on the abundances of species in those webs. Key words: consumer–resource interactions; contramensalism; hydra effect; indirect effects; interaction strength; Lotka-Volterra model; predator–prey interactions. INTRODUCTION Historically, the indirect interaction between two predators (consumers) that are limited by their intake of shared prey (resources) has been classified as competition (Gause 1934, Hutchinson 1959, Levins 1968). Competition has also been defined by its outcomes; competition involves negative effects of each species on the other’s per capita population growth rate or equilibrium population size (Schoener 1983, Abrams 1987). MacArthur’s (1970) analysis of a two-predator system in which shared prey species did not compete seemingly confirmed that each predator’s per capita growth rate and its equilibrium population size will decrease with an increase in the other’s population. Nevertheless, some theoretical studies have shown that indirect mutualistic interactions between predators are possible when shared prey species strongly compete with each other (Levine 1976, Tsumura et al. 1993, Vander- meer 2004, Abrams and Nakajima 2007). Overlap in food use by prey that share a common predator is observed in most empirically studied multitrophic-level food webs (Cohen et al. 1990, Polis 1991, Tirok and Gaedke 2010). Furthermore, empirical studies (Dodson 1970, Davidson et al. 1984, Brown et al. 1986) have shown that highly asymmetrical competition between shared prey species can produce a positive effect of one predator on another. Despite this theoretical and empirical work, models of indirect interactions between predators via their shared prey usually assume that the prey do not interact with each other (e.g., Taper and Case 1985, 1992, Abrams et al. 2008, Leimar et al. 2013). Thus, theoretical work has still not fully explored and cataloged the types of indirect interactions that are possible between predators that utilize a common set of prey. Another common assumption of resource-based competition theory is that interactions on each trophic level are symmetric. When there are only two resources (prey), symmetry of consumption implies that each consumer (predator) has its highest uptake rate on a different resource (e.g., Tsumura et al. 1993, Vander- meer 2004, Abrams and Nakajima 2007). In contrast, reviews of competition have found that asymmetric interactions predominate; an early review found evi- dence of strong asymmetry in 51 of 61 field experiments where it was examined (Schoener 1983). A later review also found high asymmetry (Denno et al. 1995). A focus on symmetry has been particularly prevalent in the few theoretical studies that have considered the possibility of Manuscript received 27 October 2014; revised 18 February 2015; accepted 6 March 2015. Corresponding Editor: B. D. Inouye. 3 Corresponding author. E-mail: michael.cortez@usu.edu 625