The previous post presented a case of apparent predator-prey interactions among ciliate species one might have expected to be competitors. There are many references to apparent interactions or indirect effects in the ecological literature (e.g., Levine 1976; Holt 1977; Lawlor 1979; Vandermeer 1980; Schaffer 1981; Bender et al. 1984). Once several ways of defining and estimating interactions from data are distinguished (see earlier post), their differences can be laid out.

**Apparent Interactions: Definitions and Estimation**

Apparent type 1 interactions should generate trajectories for the members of an apparent community that mimic the actual trajectories for those members, i.e., the trajectories generated by the dynamics of the full community. Variations of this approach depend on whether the interactions:

a. are assumed to govern trajectories near equilibrium—the method used in Taylor (2005, Chapter 1B);

b. are calculated as if the community were near equilibrium, but then extended to apply away from equilibrium by using a GLV (model 1 in previous post);

c. are derived by fitting observed trajectories directly to the GLV model; or

d. are derived by fitting observed trajectories directly to some model other than the GLV.

Other methods include a generalization of MacArthur (1972:33ff) and Schaffer’s (1981) Abstracted Growth Equations. Assuming certain special conditions, this last method can be used to derive equations for the apparent community when we do not have knowledge of the full system.

Bender et al. (1984)’s PULSE and PRESS methods: If the time scales of the hidden populations and of the modeled populations are similar, the PULSE method yields estimates of the direct interactions between the modeled populations. These are unable, in general, to mimic the actual trajectories of the modeled populations (see Taylor 2005, Chapter 1B). If the time scales are disjunct then PULSE estimates of the direct interactions may absorb effects from the hidden variables. In other words, they will be estimates of apparent interactions and thus potentially take counter-intuitive values. The PRESS method has some significant limitations. It cannot be used in most cases where direct self-interactions are zero (especially those of hidden variables), or where the apparent community has only one member. When the PRESS method can be applied, the estimated interactions are actually estimates of apparent interactions.

Type 2 interactions focus only on the two populations in question and so, in principle, are not affected by the dynamics of hidden variables. However, if loop analysis (Levins 1975) is used to calculate the values of type 2 interactions from estimates of type 1 interactions, then the full set of direct interactions must be known. (Loop analysis using apparent type 1 interactions of the form developed in Taylor 2005, Chapter 1B generate qualitatively good estimates of type 2 interactions, but calculation of such apparent interactions requires knowledge of the full set of direct interactions; Taylor 1985, 119-177.)

Adapted from Taylor, P.J. (2005) Unruly Complexity: Ecology, Interpretation, Engagement (U. Chicago Press).

**References**

Bender, E. A., T. J. Case and M. E. Gilpin (1984). “Perturbation experiments in community ecology: Theory and practice.” Ecology 65: 1-13.

Holt, R. D. (1977). “Predation, apparent competition, and the structure of prey communities.” Theoretical Population Biology 12: 197-229

Lawlor, L. R. (1979). “Direct and indirect effects of n-species competition.” Oecologia 43: 355-364

Levine, S. H. (1976). “Competitive interactions in ecosystems.” American Naturalist 110: 903-910

Levins, R. (1975). “Evolution in communities near equilibrium,” in M. L. Cody and J. M. Diamond (Eds.), Ecology and Evolution of Communities. Cambridge, MA: Harvard University Press,** **16-50.

MacArthur, R. H. (1972). Geographical Ecology. New York: Harper and Row.

Schaffer, W. M. (1981). “Ecological abstraction: The consequences of reduced dimensionality in ecological models.” Ecological monographs 51: 383-401.

Taylor, P. J. (1985). Construction and turnover of multispecies communities: A critique of approaches to ecological complexity. Unpublished Ph. D. dissertation. Cambridge, MA: Harvard University.

Vandermeer, J. H. (1980). “Indirect mutualism: variations on a theme by Stephen Levine.” American Naturalist 116: 441-448.

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