Weak trophic interactions and the balance of nature

 

KEVIN MCCANN, ALAN HASTINGS & GARY R. HUXEL

Department of Environmental Sciences and Policy, University of California, Davis, California 95616, USA

Nature 395, 794 - 798 (1998) © Macmillan Publishers Ltd.

Ecological models show that complexity usually destabilizes food webs1,2, predicting that food webs should not amass the large numbers of interacting species that are in fact found in nature3,4,5. Here, using nonlinear models, we study the influence of interaction strength (likelihood of consumption of one species by another) on food-web dynamics away from equilibrium. Consistent with previous suggestions1,6, our results show that weak to intermediate strength links are important in promoting community persistence and stability. Weak links act to dampen oscillations between consumers and resources. This tends to maintain population densities further away from zero, decreasing the statistical chance that a population will become extinct (lower population densities are more prone to such chances). Data on interaction strengths in natural food webs7,8,9,10,11 indicate that food-web interaction strengths are indeed characterized by many weak interactions and a few strong interactions.

Here we combine formally the influence of interaction strength with modern food-web data and models, uniting verbal arguments12,13,14,15,16 with the rigorous formulations of May1,2. Our analysis differs from May's contributions in five important ways. First, we use a measure of interaction strength that is based upon empirical estimates of per capita interaction strength; second, we assume that communities can display nonequilibrium dynamics; third, we construct complexity as simple food webs (after ref. 17) in a manner consistent with patterns found in nature14,15,16; fourth, we use biomass as the model currency; and fifth, we use consumption rates that become saturated as resource density increases (that is, we use type II functional responses). We describe our model and define terms in Box 1.

It is well known that the model food chain (Fig. 1a) exhibits several behaviours (such as stable equilibria, cycles, chaos and multiple attractors)18,19,20. In a simplified sense, the food chain is best understood by considering it as two coupled consumer-resource subsystems: a consumer-resource interaction (that is, the interaction between C1 and R in Equation set (1), box 1) and the top-predator-consumer interaction (that is, the interaction between P and C1 of Equation set (1)). For example, if the food chain is constructed from two strong consumer-resource interactions (that is, both subsystems produce cyclic behaviour) then the food chain behaviour becomes quite complex and variable18,19,20. In this case, the food chain can be seen two coupled oscillators whose dynamics depend on whether the frequencies of these oscillators are commensurate (producing cyclic dynamics) or incommensurate (producing quasi-periodic or chaotic dynamics).

Figure 1 The six food-web configurations studied are: a, a simple food chain, b, a food web with multiple intermediate consumers (exploitative competition), c, a food web with the top predator feeding on two intermediate consumers (apparent competition), d, a food web with consumer 1 feeding on the basal resource and on the second consumer (intraguild predation), e, a food chain including omnivory, and f, a food chain with external inputs. R denotes the basal resource species; C1 and C2 denote intermediate consumer species 1 and 2; and P denotes the top predator ( Box 1).

Two corollaries follow from these statements: first, stabilizing all the underlying oscillators eliminates the occurrence of cyclic or chaotic dynamics in the full system; and second, reducing the amplitude of the underlying oscillators reduces the amplitude of the dynamics of the full system. Thus we predict that inhibiting strong consumer-resource interactions within a food web promotes persistence in food webs.

 

We study below how interaction strength influences oscillatory subsystems of more complicated food webs (Fig. 1b-f). We show that weak interactions can act to inhibit potentially oscillatory subsystems through the following three naturally occurring mechanisms.

In the apparent competition mechanism, apparent competition among resource species occurs when a consumer preys upon multiple resources. In this case, a consumer can inhibit a potentially oscillatory consumer-resource interaction when it trades off preference for one resource (that is, that resource involved in the potentially oscillatory consumer-resource interaction) in order to feed on a second resource. This effectively reduces the efficiency at which the consumer attacks the first resource.

In the exploitative competition mechanism, two consumers compete for the same resource. In this case, the addition of a second competitor reduces the growth rate of the shared resource item (from the perspective of the first consumer) and, therefore, can inhibit a potentially oscillatory consumer-resource interaction involving the first consumer.

In the food-chain-predation mechanism, food-chain predation occurs when a top predator feeds on an intermediate consumer which feeds on a resource. The top predator can inhibit the growth rate of its resource (the intermediate consumer) and, therefore, inhibit the intermediate-consumer-resource trophic interaction. The top predator reduces the intermediate consumer's attack rate on the resource item.

At the heart of these mechanisms is the concept that a stable consumer-resource interaction is required to dampen the dynamic behaviour of a potentially strong interaction (and, hence, a potentially oscillatory interaction). Consistent with the theory of consumer-resource interactions, reduction in growth rates of the resource and reduction in attack rates by the consumer tend to stabilize a consumer-resource interaction21.

We now blend more food-web structure into the simple food-chain model (Fig. 1a), enabling us to study the dynamic implications of naturally occurring food-web structures4,14,15,16. Looking at exploitative competition22 (Fig. 1b), apparent competition23 (Fig. 1c), and intraguild predation24 in which consumer 1 feeds on both the basal resource and a second consumer (Fig. 1d), we discuss how the three inhibiting mechanisms naturally arise and work to stabilize food-web dynamics. We then discuss how these mechanisms also arose in previous studies in which it was found that weak amounts of omnivory22,25,26 (Fig. 1e) and external (allochthonous) inputs4,27 (Fig. 1f) can enhance food-web stability.

To study exploitative competition, we began with parameters for a biologically plausible example of persistent chaotic dynamics for a three-species food chain (Omegaij values = 1; xC1 = 0.40; yC1 = 2.009; xP = 0.08; yP = 5.0; C0 = 0.50; and Ro = 0.16129; see Box 1 for definitions)19. Figure 1b shows the addition to this chain of a second intermediate consumer (C2), which interacts with the basal resource, R, with a strength governed by the parameter OmegaC2R. In this case, Equation set (1) has the following extra parameters: xC2 = 0.20; yC2 = 3.50; R02 = 0.90. We chose the new consumer, C2, to be competitively inferior to C1, so its ability to persist is mediated by the selective predation of the top predator, P, on C1. Figure 2a depicts the local minima and maxima attained for the top predator densities, P, in solutions to Equation set (1) across a range of C2-R interaction strengths relative to C1-R interaction strengths (that is, IC2R/IC1R). Figure 2a also shows food-web diagrams that depict the change in food-web structure as the relative interaction strength changes value.

The local minima and maxima for top predator density, P, attained in the attracting solutions for a range of relative interaction strengths. Food-web configurations are given as a function of the relative interaction strengths. Food-web configurations are given as a function of the relative interaction strength. Whenever the configuration lacks an explicit link between a species and the rest of the connected web this implies that the species cannot persist. a, Exploitative competition. b, Apparent competition. c, Intraguild predation. d, The configuration used in b, starting with a limit cycle solution (IC2R/IC1R = 0.62).

In this scenario we expect the exploitative competition mechanism to inhibit the oscillating C1-R subsystem. It does exactly this when C2 can invade (IC2R/IC1R 0.102). Below this value the original food chain (P-C1-R) remains intact and continues to exhibit chaotic dynamics (shown by the thick set of points from 0.50 to 0.57 in Fig. 2a). However, when C2 can invade, the dynamics immediately begin to take on a much simpler periodic signal that tends further away from zero (0.102-0.125). Equation set (1) does not reach an equilibrium solution over this range, as the P-C1 oscillator remains intact--none of the mechanisms is operating to inhibit this oscillatory component.

 

When the relative interaction strength increases beyond 0.125, the attractor begins to become less bounded again and eventually undergoes a period-doubling cascade to a more complex dynamical regime above IC2R/IC1R 0.15. The new C2-R interaction has become too strong (and therefore oscillatory) and no longer dampens the system; it contributes to the chaos by adding a third oscillating subsystem to the food web.

To study apparent competition, we began with the same parameters as above except that we let IC2R/IC1R 0.154 (that is, OmegaC2R = 0.98). This returns us to a complex oscillatory food-web dynamic (Fig. 2a); however, in this case, we have three oscillatory subsystems (P-C1, C1-R, and C2-R). We now construct a link between the top predator, P, and the second intermediate consumer, C2 (Fig. 1c), by allowing OmegaPC1 to be <1, creating apparent competition between the two intermediate consumers.

Figure 2b shows the local minima and maxima for the top predator, P, in solutions to Equation set (1) across a range of strengths of the P-C2 interaction relative to strengths of the P-C1 interaction (IPC2/IPC1). We expect the apparent competition mechanism to inhibit the C2-R and C1-R oscillators. As relative interaction strength increases from zero, it immediately causes a period-doubling reversal, forcing simpler, more bounded, limit cycle dynamics for IPC2/IPC1 0.03-0.10 and stable equilibrium dynamics for IPC2/IPC1 0.10-0.12. As before, increasing the strength of the interaction further eventually destabilizes the system. Despite the complexity of this system, which includes multiple attractors and numerous bifurcations, qualitatively the result remains: relatively weak links simplify and bound the dynamics of food webs.

We now construct a link such that C1 can feed on C2 (Fig. 1d), creating intraguild predation within the food web. The parameters are the same as in the previous case, with IPC2/IPC1 0.01; (that is, OmegaPC1 = 0.99) so that we start off with a complex oscillatory dynamic (we have three oscillators in the food web again). Figure 2c shows the local minima and maxima for the top predator, P, in solutions to Equation set (1) across a range of the relative interaction strength IC1C2/IC1R). Here, we expect the apparent competition mechanism to inhibit the C1-R subsystem and we expect the food-chain mechanism to inhibit the C2-R subsystem. As a result of having two inhibitors and three potential oscillators, the dynamics never reach a locally stable equilibrium; they attain a well-bounded limit cycle solution (driven by the remaining oscillating P-C1 subsystem) for most IC1C2/IC1R values above 0.05.

In two of the cases above, weak links failed to beget stable local equilibria with all species present--at best we get well-bounded limit cycle solutions. This is because, in these two cases, there is not an inhibiting mechanism for each potential oscillator. This is largely a result of our choice of starting with chaotic dynamics and adding only one additional food-web interaction at a time: our analysis biases our results to have fewer inhibitors than oscillators. Finally we show that adding another, appropriately directed, inhibiting mechanism to such a situation allows for rapid local stabilization to an equilibrium solution. Figure 2d depicts solutions to system 1 when we start off with IC2R/IC1R = 0.11 (Fig. 2a). Thus, we are starting off with one oscillating subsystem (the P-C1 subsystem). After adding the apparent competition mechanism, which inhibits the P-C1 subsystem, we rapidly get a locally stable solution for weak relative interaction strengths (by IPC2/IPC1 0.040 in Fig. 2d).

Using a model formulation similar to system 1, two groups26,27 have shown that weak amounts of omnivory (Fig. 1e) and allochthonous inputs (Fig. 1f) bound the magnitude of the oscillations in P-C1-R densities further away from zero. In both cases, chaotic dynamics collapsed through period-doubling reversals (that is, the periodicity of solutions reduced from n to n/2 to n/4 etc.) into well-bounded cyclic or stable dynamics under weak to moderate amounts of omnivory and allochthonous inputs. Both results can be explained with our proposed mechanisms.

Several testable predictions for food webs result from our study. First, if a relatively weak interaction exists for each strong consumer-resource interaction, then the food web should be stabilized relative to the oscillatory subsystems (that is, the food web should be less oscillatory). Second, if food webs have many weak interactions, then, at least form a deterministic viewpoint, chaotic dynamics are unlikely. Third, generalist-dominated food webs should exhibit less variable dynamics than specialist-dominated food webs. Fourth, depauperate food webs should tend to be more oscillatory than reticulate food webs as depauperate food web species tend to have larger average interaction strengths, thus promoting the dominance of a few strong (oscillatory) interactions. Finally, if we assume the realizable interaction scope is proportional to interaction scope, then given all else equal, endotherms (ymax,i = 1.60) and vertebrate ectotherms (ymax,i = 3.90) are more likely to be stabilized by weak food web links than invertebrates as invertebrates have a greater interaction scope (since ymax,i = 19.4) and thus greater potential to maintain a larger number of strong consumer-resource interactions. A larger number of strong consumer-resource interactions requires a greater number of weak interactions to inhibit oscillatory subsystems. It follows, given topologically identical food webs, that invertebrate-dominated communities are more likely to have the most oscillatory dynamics. Our overall conclusion is that knowledge of interaction strength in study of food webs is vital. Although few quantitative field estimates of interaction strength are available, early data unequivocally indicate that distributions of interaction strength are strongly skewed towards weak interactions7,8,9,10,11. It seems, then, that weak interactions may be the glue that binds natural communities together.

Received 17 June;accepted 15 August 1998.

------------------

References

  1. May, R. M. Stability in multi-species community models. Math. Biosci. 12, 59-79 (1971). Links Save Citation
  2. May, R. M. Stability and Complexity in Model Ecosystems (Princeton Univ. Press, Princeton, 1973). Links Save Citation
  3. Winemiller, K. O. Spatial and temporal variation in tropical fish trophic networks. Ecol. Monogr. 60, 331-367 (1990). Links Save Citation
  4. Polis, G. A. Complex trophic interactions in deserts: an empirical critique of food web theory. Am. Nat. 138, 123-155 (1991). Links Save Citation
  5. Lavigne, D. M. in Studies of High Latitude Homeotherms in Cold Ocean Systems (ed. Montevecchi, W. A.) 95-102 (Canadian Wildlife Service Occasional Paper, Ottawa, Canada, 1995). Links Save Citation
  6. Gardner, M. & Ashby, W. R. Connectance of large dynamical (cybernetic) systems: critical value for stability. Nature 228, 784 (1970). Links Save Citation
  7. Paine, R. T. Food-web analysis through field measurement of per capita interaction strength. Nature 355, 73-75 (1992). Links Save Citation
  8. Fagan, W. F. & Hurd, L. E. Hatch density variation of a generalist athropod predator: population consequences and community impact. Ecology 75, 2022-2032 (1994). Links Save Citation
  9. Wootton, J. T. Estimates and test of per capita interaction strength: diet, abundance, and impact of intertidally foraging birds. Ecol. Monogr. 67, 45-64 (1997). Links Save Citation
  10. Goldwasser, L. & Roughgarden, J. Construction and analysis of a large caribbean food web. Ecology 74, 1216-1233 (1993). Links Save Citation
  11. Rafaeli, D. G. & Hall, S. J. in Food Webs: Integration of Patterns & Dynamics (eds Polis, G. A. & Winemiller, K. O.) 185-191 (Chapman & Hall, New York, 1996). Links Save Citation
  12. MacArthur, R. H. Fluctuations of animal populations and a measure of community stability. Ecology 36, 533-536 (1955). Links Save Citation
  13. Elton, C. S. Ecology of Invasions by Animals and Plants (Chapman & Hall, London, 1958). Links Save Citation
  14. Strong, D. Are trophic cascades all wet? Differentiation and donor-control in speciose ecosystems. Ecology 73, 747-754 (1992). Links Save Citation
  15. Polis, G. A. Food webs, trophic cascades and community structure. Aust. J. Ecol. 19, 121-136 (1994). Links Save Citation
  16. Polis, G. A. & Strong, D. Food web complexity and community dynamics. Am. Nat. 147, 813-846 (1996). Links Save Citation
  17. Holt, R. D. in Multitrophic Interactions (eds Begon, M., Gange, A. & Brown, V.) 333-350 (Chapman & Hall, London, 1996). Links Save Citation
  18. Hastings, A. & Powell, T. Chaos in a three species food chain model. Ecology 72, 896-903 (1991). Links Save Citation
  19. McCann, K. & Yodzis, P. Biological conditions for chaos in a three-species food chain. Ecology 75, 561-564 (1995). Links Save Citation
  20. De Feo, O. & Rinaldi, S. Singular homoclinic bifurcations in tritrophic food chains. Math. Biosci. 148, 7-20 (1998). Links Save Citation
  21. Rosenzweig, M. & McArthur, R. H. Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 107, 275-294 (1963). Links Save Citation
  22. Diehl, S. Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships. Oikos 68, 151-157 (1993). Links Save Citation
  23. Holt, R. D. Predation, apparent competition, and the structure of prey communities. Theor. Popul. Biol. 12, 197-229 (1977). Links Save Citation
  24. Polis, G. A. & Holt, R. D. Intraguild predation: the dynamics of complex trophic interactions. Tree 7, 151-155 (1992). Links Save Citation
  25. Fagan, W. F. Omnivory as a stabilizing feature of natural communities. Am. Nat. 150, 554-567 (1997). Links Save Citation
  26. McCann, K. & Hastings, A. Re-evaluating the omnivory-stability relationship in food webs. Proc. R. Soc. Lond. B 264, 1249-1254 (1997). Links Save Citation
  27. Huxel, G. & McCann, K. Food web stability: the influence of trophic flows across habitats. Am. Nat. 152, 460-469 (1998). Links Save Citation
  28. Yodzis, P. & Innes, S. Body-size and consumer-resource dynamics. Am. Nat. 139, 1151-1175 (1992). Links Save Citation
  29. Chesson, J. The estimation and analysis of preference and its relationship to foraging models. Ecology 64, 1297-1304 (1983). Links Save Citation

Acknowledgements. We thank P. DeValpine, M. Hoopes, D. Strong, P. Yodzis and J. Paloheimo for discussions; D. Post, M. E. Conners and D. S. Goldberg for a preprint of a related manuscript; and the US National Science Foundation and the Institute of Theoretical Dynamics for their support.

 

Correspondence and requests for materials should be addressed to K.M. (e-mail: kevin@six.ucdavis.edu).