Bulletin of Mathematical Biology 2007 69: 2093 2104 DOI 10.1007/s11538-007-9214-0 ORIGINAL ARTICLE Antagonistic and Synergistic Interactions Among Predators Gary R. Huxel Department of Biological Sciences, University of Arkansas, Fayetteville, AR 72701, USA Received: 28 April 2006 / Accepted: 7 March 2007 / Published online: 7 June 2007 Society for Mathematical Biology 2007 Abstract The structure and dynamics of food webs are largely dependent upon interactions among consumers and their resources. However, interspecific interactions such as intraguild predation and interference competition can also play a significant role in the stability of communities. The role of antagonistic/synergistic interactions among predators has been largely ignored in food web theory. These mechanisms influence predation rates, which is one of the key factors regulating food web structure and dynamics, thus ignoring them can potentially limit understanding of food webs. Using nonlinear models, it is shown that critical aspects of multiple predator food web dynamics are antagonistic/synergistic interactions among predators. The influence of antagonistic/synergistic interactions on coexistence of predators depended largely upon the parameter set used and the degree of feeding niche differentiation. In all cases when there was no effect of antagonism or synergism a ij = 1.00, the predators coexisted. Using the stable parameter set, coexistence occurred across the range of antagonism/synergism used. However, using the chaotic parameter strong antagonism resulted in the extinction of one or both species, while strong synergism tended to coexistence. Whereas using the limit cycle parameter set, coexistence was strongly dependent on the degree of feeding niche overlap. Additionally increasing the degree of feeding specialization of the predators on the two prey species increased the amount of parameter space in which coexistence of the two predators occurred. Bifurcation analyses supported the general pattern of increased stability when the predator interaction was synergistic and decreased stability when it was antagonistic. Thus, synergistic interactions should be more common than antagonistic interactions in ecological systems. Keywords Synergistic and antagonistic interactions Predator predator interactions Food webs Trait-mediated effects 1. Introduction Interspecific interactions among predators are known to influence the structure and stability of food webs Elton, 1958; May, 1973; Polis et al., 1989. Predator predator interactions can influence the total predation rate of a community of predators resulting in E-mail address: ghuxel@uark.edu.
2094 G.R. Huxel cascading effects on prey populations Sih et al., 1985, 1998; Wilbur and Fauth, 1990; Polis and Holt, 1992; Schroeder, 1996. Several outcomes are possible given predator predator interactions: 1 the predators can act synergistically resulting in greater than expected rates of predation if considering each predator independently synergism; 2 total mortality is equivalent to the expected value additive mortality; and 3 total mortality is less than expected nonadditive mortality. Nonadditive mortality can arise from several different mechanisms including intraguild predation, interspecific competition, and antagonistic predator predator interactions. The first two have been drawn significant attention in theoretical and empirical studies, but the last has been largely ignored. This is mainly due to the problem of distinguishing among these interactions. In antagonistic interactions, reduced capture rates can arise due to behavioral changes in the prey in response to one or more predators Sih et al., 1985, 1998; Bjorkman and Liman, 2005. This may also arise in cases where predators avoid one another through fine scale habitat selection or to lower risk of intraguild predation. The effect of microscale habitat selection is not addressed here. Synergism occurs when these interactions result in increased prey capture rates relative to when only one predator species is present. The attempt by prey to flee one predator may actually result in greater predation risk from a second predator McPeek, 1990. For example, the response of mayflies to predatory stoneflies is to move into the water column where risk of predation by trout is increased Cowan and Peckarsky, 1994. Thus, both antagonism and synergism directly influence the predators and their prey species through prey capture rates. Antagonism has a negative effect on predator population growth and a positive affect on prey population growth, while synergism has the opposite effect. Further, indirect effects due to antagonism and synergism can potentially dominate the dynamics of the food web Wootton, 1994; Polis and Strong, 1996. Simulations and bifurcation analyses are used to explore the overall influence of antagonism and synergism on food web stability. Given that predators influence directly or indirectly the ability of others to capture prey, what are the consequences of antagonistic and synergistic interactions? Do their interactions increase or decrease stability in terms of coexistence of the predators and stability of population dynamics? It is expected that synergistic interactions will result in more efficient predation, thus lowering prey densities. If the prey species can persist, then the long-term effect would be to lower predator growth rates stabilizing the food web, however if the prey become extinct so will the predator. If predation rates are too high, prey populations followed by predator populations will become extinct. Similarly, if the interactions are antagonistic, then predator growth rates are predicted to be decreased due to lower per capita predation rates. This may allow for increased prey densities. Coexistence of the predators may occur even if antagonism greatly reduced per capita capture rates given sufficiently large prey populations. However, increased prey densities may result in resource depletion and coupled with increased predation with a lagged response with predators slowly tracking prey densities may result in strong oscillations in prey and predator densities with either one or the other becoming extinct. While the focus of this work is on predator predator interactions, prey species may respond to predators through morphological, behavioral, or life-history traits. These traitmediated interactions have been shown to strongly influence food web dynamics including the strength of trophic cascades Schmitz et al., 2004; Bolker et al., 2003. Thus, while
Antagonistic and Synergistic Interactions Among Predators 2095 trait-mediated interactions are not directly addressed, they are will influence whether antagonistic/synergistic interactions occur and the strength of those interactions. Using a five species food web model, direct and indirect effects of antagonistic/synergistic interactions on food web stability was investigated. In particular, the interactions predator-prey interaction strengths, diet overlap and antagonistic/synergistic interactions and their influences on predator coexistence and population dynamics were examined using simulations and bifurcation analyses of a tritrophic model system. 2. Methods 2.1. Model systems The basic model structure is based upon previous studies of a well-known set of predatorprey equations Hastings and Powell, 1991; Yodzis and Innes, 1992; McCann et al., 1998; Huxel and McCann, 1998; Huxel et al., 2002. The Yodzis and Innes 1992 parameterization of the Hastings and Powell 1991 tritrophic food chain model was the starting point of all model systems and is given below: dr = R 1 R HR x H y H, dt K R + R 0 dh HR = x H H + x H y H, dt R + R 0 where consumer-resource interactions exhibit Type II functional responses; R is the resource species; K is the carrying capacity of R; H is the consumer; R 0 is the half saturation point for the functional response between the consumer and resource levels; x i is the metabolic rate of trophic level i, measured relative to the production-to-biomass ratio of the resource density; y i is a measure of the ingestion rate per unit metabolic rate of the resource by i. This basic model can exhibit a wide range of dynamics from chaos to limit cycles to stable points depending upon the parameter set used Yodzis and Innes, 1992; McCann and Yodzis, 1994. Given this range of dynamics, various theoretical studies have explored factors that can stabilize the dynamics of food webs using this basic model to represent predator-prey interactions for example: McCann and Hastings, 1997; McCann et al., 1998; Huxel and McCann, 1998. This work expands on those studies by examining the influence of antagonistic/synergistic interactions among predators of food webs in which the predator-prey interactions are based upon equations in 1. Increasing food web complexity via the addition of a second prey can increase stability of the food web given that the new predator-prey interactions are weak to moderate in strength McCann et al., 1998. Thus, a model with two prey and two predators was examined to determine whether the influence of antagonistic/synergistic interactions changes with food web structure. With multiple prey, specialization on a particular prey species becomes possible and can reduce resource competition without influencing the antagonistic/synergistic interactions. To examine the potential influence of specialization on 1
2096 G.R. Huxel the impact of antagonistic/synergistic interactions on food web stability a system of five species, now with two prey species, is utilized. This system is given by: dr = R 1 R RH1 RH2 x c y c x c y c, dt K R + R 0 R + R 0 dh 1 RH1 H 1 P 1 = x c H 1 + x c y c α 45 ω 24 x p y p dt R + R 0 ω 24 H 1 + ω 34 H 2 + H 0 H 1 P 2 α 54 ω 25 x p y p, ω 25 H 1 + ω 35 H 2 + H 0 dh 2 RH2 H 2 P 1 = x c H 2 + x c y c α 45 ω 34 x p y p dt R + R 0 ω 24 H 1 + ω 34 H 2 + H 0 H 2 P 2 α 54 ω 35 x p y p, 2 ω 25 H 1 + ω 35 H 2 + H 0 dp 1 H 1 P 1 = x p P 1 + α 45 ω 24 x p y p dt ω 24 H 1 + ω 34 H 2 + H 0 H 2 P 1 + α 45 ω 34 x p y p, ω 24 H 1 + ω 34 H 2 + H 0 dp 2 H 1 P 2 = x p P 2 + α 54 ω 25 x p y p dt ω 25 H 1 + ω 35 H 2 + H 0 H 2 P 2 + α 54 ω 35 x p y p, ω 25 H 1 + ω 35 H 2 + H 0 where ω ij s are the preference coefficients indicating the degree of specialization for prey i by predator j.theω ij s sum to 1.0 for each predator e.g. ω 25 + ω 25 = 1. By varying the ω ij s the amount of diet overlap can be controlled. Analytical analysis of the stability of this system is complex, so numerical simulations of the model systems were used to explore the influence of antagonism/synergism on stability of the above food web 2. The a ij s are not density dependent in this formulation. This is for two reasons, one is mathematical simplicity and the other is to reflect the biology of prey responses to density independent cues. In future studies, the density dependence of the antagonistic/synergistic response should be addressed. 2.2. Simulation methods The effect of interactions among multiple predators and their prey maybe confounded by increased density when a second predator is introduced. In order to minimize this effect, total initial densities of predators were kept constant so that each predator in a two predator system had an initial density of half the predator in a one predator system. Parameter values used in the simulations were chosen so that each predator-prey pair individually displayed either chaotic dynamics: x H = 0.4, y H = 2.009, R 0 = 0.16129, x P = 0.08, y P = 5, H 0 = 0.5; stable dynamics x H = 0.4, y H = 2.009, R 0 = 0.5, x P = 0.01, y P = 5, H 0 = 1.5; or a limit cycle x H = 0.4, y H = 2.009, R 0 = 0.3333, x P = 0.5, y P = 5, H 0 = 0.5 Yodzis and Innes, 1992; McCann and Yodzis, 1994; Huxel and McCann, 1998. Simulations were performed over a range of antagonistic/synergistic coefficients
Antagonistic and Synergistic Interactions Among Predators 2097 from weak antagonism α ij s = 0.50 to no antagonism/synergism α ij s = 1.00 to strong synergism a ij s = 1.50. In the two prey model simulations, simulations were performed for generalist predators ω ij s = 0.50 and specialist predators ω ij s = 0.90 for the primary prey and ω ij s = 0.10 for the secondary prey but each predator specialized on a different prey. All simulations were performed for 1,000 time steps to examine the influence of antagonism/synergism on stability in terms of population dynamics of predators and prey. 2.3. Bifurcation analyses Bifurcation analyses were performed setting the α ij s of the two predators equal over the range of 0.50 to 1.50 so that these analyses were performed on cased of symmetric interactions. Analyses were done for both generalist predators and specialist predators. The minima and maxima densities are taken over a 500 time step period after an initial 500 time steps. 3. Results 3.1. Antagonism/synergism Coexistence of predators was greatly influenced by the level of antagonism/synergism, degree of diet specialization, and the parameter set used Figs. 1, 2. For the chaotic parameter set, the degree of diet specialization interacted with the strength of the antagonism/synergism to produced distinct patterns of coexistence Fig. 1. Neither predator persisted under antagonistic interactions a ij s < 1.00. For generalist predators, coexistence only occurred under conditions when the a ij s were synergism and both were relatively large Fig. 1A. For specialist predators the results were more variable in part due to the chaotic dynamics resulting in some predator populations becoming extinct due to low population numbers. But, again persistence and coexistence occurred only under synergistic conditions Fig. 1B. Neither predator persisted in both the generalist and specialist simulations at the null value of a ij s = 1.00 no effect of antagonism or synergism. For the limit cycle parameter set Fig. 2, there were significant differences in coexistence between the generalist and specialist simulations. For generalist predators, coexistence occurred only when the antagonism/synergism coefficients where symmetrical Fig. 2A. For specialist predators, coexistence was much more likely and occurred a large range of a ij values Fig. 2B. The predators coexisted in both the generalist and specialist simulations at the null value of a ij s = 1.00. For the stable parameter set, coexistence occurred over all of the antagonism/synergism parameter space. This allows for an examination of another factor of stability minimum population size under conditions when coexistence always occurred asking whether minimum population size increase with antagonism, synergism or both? 3.2. Bifurcation analyses The bifurcation plots for the chaotic parameter set show that both the minima and maxima densities sharply increase at α ij = 1.14 Fig. 3. As the α ij s increase above this value,
2098 G.R. Huxel A B Fig. 1 Regions of extinction and coexistence in the two prey two predator antagonism/synergism model simulations using the chaotic parameter set. α ij s are the antagonism/synergism coefficients. A Generalist predators each predator has a 0.50 preference for each prey species. B Specialist predators each predator has a 0.90 preference for different prey species. Dark blue designates extinction of both predators; light blue = predator 2 only; yellow = predator 1 only; and red = coexistence of both predators.
Antagonistic and Synergistic Interactions Among Predators 2099 A B Fig. 2 Regions of extinction and coexistence in the two prey two predator antagonism/synergism model simulations using the limit cycle parameter set. α ij s are the antagonism/synergism coefficients. A Generalist predators. B Specialist predators. Color scheme same as in Fig. 1.
2100 G.R. Huxel A B Fig. 3 Bifurcation chaotic parameter set. A General predators. B Specialist predators. the minima and maxima slightly decrease. The pattern was similar for both the generalist and specialist analyses. The dynamics also undergo a shift moving from chaotic dynamics when coexistence does not occur to a limit cycle when coexistence does occur. For the limit cycle parameter set, coexistence occurs under relatively strong antagonistic interactions a ij = 0.56 Fig. 4. From that point to a ij = 0.84 a stable equilibrium is exhibit with a slight decrease. Above a ij = 0.84 the system moves into a limit cycle with the maxima increase and the minima decrease with a ij. The generalist and specialist are similar, except at high a ij s when in the specialist analyses the system moves into a chaotic regime Fig. 4B. For the stable parameter set, coexistence at a stable equilibrium occurred for all values of a ij Fig. 5. The minimum and maximum values however are very low until a ij > 1. The population size exhibits a shift first at a ij = 1.00 and then increased rapidly above
Antagonistic and Synergistic Interactions Among Predators 2101 A B Fig. 4 Bifurcation limit cycle parameter set. A General predators. B Specialist predators. a ij 1.20 suggesting that synergism has a strong stabilizing effect. The pattern is similar for both the generalist and specialist predators. 4. Discussion Predator predator interactions can play a strong role in structuring food webs. The interactions among multiple predators can produce predation rates that deviate from the expected additive effect of their individual impacts on either prey Sih et al., 1998. The effect can be positive antagonistic predator interactions or negative synergistic predator interactions on prey populations. However, indirect effects can cascade through the food web and can potentially have a stronger effect that direct effects Wootton, 1994;
2102 G.R. Huxel A B Fig. 5 Bifurcation stable parameter set. A General predators. B Specialist predators. Polis and Strong, 1996. Thus, predicting the net outcome of antagonistic/synergistic interactions on predator populations can be difficult, but it has important consequences for management of both predator and prey populations. The results demonstrate that coexistence of the two predators occurs mostly under conditions of synergistic interactions; however, the degree of coexistence varies with parameter set, diet overlap, and the strength of the antagonistic/synergistic interactions. Using the two oscillating parameter sets chaotic and limit cycle, antagonistic interactions resulted in increased prey populations, however, the prey populations would crash due to resource depletion and a lagged response of predators to prey densities. This would strongly drive down the prey populations and either the prey and predators or the predators would become extinct as a result similar to the dynamics of a strongly oscillating Lotka Volterra predator-prey model. With synergistic interactions, prey densities are lower but preda-
Antagonistic and Synergistic Interactions Among Predators 2103 tors are more effective resulting in more stable dynamics as exhibited in the bifurcation diagrams Figs. 3, 4. Minimum densities increased sharply as the a ij s moved towards and into values indicative of synergism. Using the stable parameter set, the equilibrium densities increased with a ij suggesting a strong stabilizing effect Fig. 5. The results also depended upon the degree of diet overlap of the predators. In the generalist case, diets were the same. In the oscillating parameter sets, diet overlap decreased stability whereas diet separation increased stability as would be expected Wilson and Yoshimura, 1994; Abrams, 1999. However, the bifurcation diagram for the limit cycle parameter set showed some unexpected results. In the specialist case, as synergistic interactions increased above a ij = 1.33, the system moved into a chaotic regime with large oscillations Fig. 4B. This did not occur in the generalist case. Thus this study, synergistic interactions actually increased coexistence and stability via reducing prey populations, thereby slowing the overall dynamics of the system. Stability as measured by coexistence and minimum population size increased with synergism strength. It appears that as predator populations increased prey populations decreased, followed by a decrease in predator populations. At low predator and prey populations the synergistic interactions allowed them to maintain positive but low growth rates. This allows for increased densities of the basal species and high reproductive rates of the prey most of which were then predated. These results also suggest that strong antagonistic interactions should be relatively rare or have some mechanism such as fine scale habitat separation or reduced diet overlap to minimize the antagonistic interaction. Another outcome is that if prey capture rates become too low, one predator may switch to feeding upon another predator resulting in intraguild predation or cannibalism. A major assumption of intraguild predation is that the intraguild prey the intermediate predator is more easily captured by the intraguild predator top predator than the shared prey species. Thus, one would expect that purely antagonistic interactions will be rare in nature relative to purely synergistic interactions and that intraguild predation should arise when predators negatively influence prey capture rates of other predators. The influence on prey and basal resource populations and the strength of trophic cascades will depend strongly on whether predator predator interactions are synergistic or antagonistic including intraguild predation. The role of trait-mediated interactions in determining whether antagonistic or synergistic interactions will also need to be further explored. Schmitz et al. 2004 have shown that trait-mediated interactions are common in ecological systems and thus it is expected that antagonistic/synergistic interactions that may arise from those interactions may play an important role in food web structure and dynamics. Acknowledgements I thank Matt Dekar, Drew Talley, and various anonymous reviewers for comments that improved this paper. This paper was supported in parts by grants DEB-0079426 and BE: CBC-0221834 from the National Science Foundation. References Abrams, P.A., 1999. The adaptive dynamics of consumer choice. Am. Nat. 153, 83 97. Bjorkman, C., Liman, A.S., 2005. Foraging behaviour influences the outcome of predator predator interactions. Ecol. Entomol. 30, 164 169.
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