Earlier posts distinguished several ways of defining and estimating interactions from data, presented a a case of apparent predator-prey interactions among ciliate species one might have expected to be competitors, and laid out differences among the many references to apparent interactions or indirect effects in the ecological literature. This post examines Levins’s loop analysis against this background.
Suppose that an ecological community has a feasible [non-negative] equilibrium, but that the equilibrium population sizes readjust to new values as the conditions under which the community operates change. If ecologists assume that their observations of population sizes at different times are actually observations of population equilibria under different conditions, they can define the effect of one population on another in terms of their relative equilibrium values under the changing conditions—do they go up or down together, in different directions, or remain unchanged? This relation between two populations combines direct and indirect effects, because it builds in the interactions among all the populations, not only the two in focus.
Levins’ loop analysis is the earliest and most general method of using the terms of the following model to calculate such effects.
Model: Near equilibrium linear form
Rate of change of population X =
Self-interaction within the X’s +
Sum of interactions of each of the other populations on X;
where the first term is a constant times the deviation of population X from its equilibrium value, and the inter-population terms are constants times the deviations of the other populations from their equilibrium values.
Levins’s method is as follows. Suppose the change in conditions of the community—or, more generally, of any dynamical system—can be expressed as a change in some parameter, C, that directly affects the rate of change of one of the populations, X. The change in the equilibrium values of each of the populations can be calculated using a complicated expression involving all the interaction and self-interaction terms in Model 2 and the partial derivative of the rate of change of X with respect to C (Levins 1975, 40). This can be calculated exactly, even though loop analysis customarily uses the sign of the interaction terms only and generates qualitative predictions (which may be indeterminate). Whether the two equilibrium populations change in the same or in different directions depends on the “node”—the population directly affected by a change in C.
In the context of a discussion of apparent interactions, the relevant question is whether hidden variables confound the values derived in loop analysis. This can be ascertained by calculating the changes in equilibrium populations using a sub-community only and comparing the results with those calculated using the full community. I investigated this question using the 5 population consumer sub-community of the 8-species web below (Taylor 1985, 150-153).
First, I generated loop analysis results using the direct interaction values for the consumer sub-community. Unfortunately these were meaningless, because the predicted local stability was the opposite of what was correct. Hoping to overcome this problem, I followed conventional loop analysis protocol and added qualitative self-inhibitory values to represent the hidden resources (Lane and Levins 1977). The predicted changes were indeterminate in every case. Finally, I used the apparent interaction values summarized in the figure below (see Taylor 1985 for calculation of the values). Although the quantitative agreement was poor, the qualitative agreement was moderately good (see Table).
Table. Qualitative loop analysis predictions for the five species consumer sub-community
Predicted change in equilibrium value of population size of species
Node of change in C |
2 |
4 |
5 |
6 |
8 |
2 |
+ |
+ |
-(+) |
– |
– |
4 |
+ |
+ |
– |
+ |
+(-) |
5 |
– |
– |
+ |
+ |
+ |
6 |
+ |
+(-) |
– |
+ |
– |
8 |
– |
– |
+ |
+ |
+ |
(Figures in parentheses indicate predictions calculated using apparent interactions that differ from the correct predictions calculated using the complete set of interactions.)
Further investigation of the effect of hidden variables on loop analysis is needed, but these initial results suggest that apparent interactions might give qualitatively reliable loop analysis predictions, while direct interactions, even if supplemented by self-inhibitory interactions, do not. The loop analyst would need the direct interactions for the complete community or be able to substitute apparent interactions of the correct sign. But, as shown in an earlier post, the sign of apparent interactions is not necessarily intuitive. What is to be done?
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