Theoretical concepts and empirical approaches to measuring interaction strength

Ecology, March, 1998 by Mark S. Laska, J. Timothy Wootton

INTRODUCTION

One of the central themes in community ecology is the examination of interactions between species (e.g., Connell 1983, Schoener 1983, Ricklefs 1990). Most pairwise interactions, and those embedded in complex food webs, are described in terms of the concept of interaction strength. Interaction strength estimates the magnitude of the effect of one species on another, and is commonly used in both experimental and theoretical approaches. This widely used concept, however, has several different definitions (e.g., MacArthur 1972, Vandermeer 1972, May 1973, Yodzis 1989), and a surprising variety of empirical approaches have been applied to determine its magnitude (e.g., Paine 1980, 1992, Davidson 1980, Bender et al. 1984, Wootton 1993a, b, 1994b, 1997, Menge et al. 1994, Laska 1995, unpublished manuscript, Moore et al. 1996; J. Ruesink, unpublished manuscript).

The great diversity of approaches and definitions complicates coherent and universal analysis of interactions in food webs. It is important for ecologists to understand interaction strength, because of its dominant role as a descriptor of community parameters (Paine 1992, Wootton 1994b, Laska 1995), because it is a critical parameter in many food web models (Pimm 1982, Lawton 1989, Pimm et al. 1991, Lawton 1992), and because it has widespread implications for conservation strategies (Mills et al. 1993, Power et al. 1996). Nonetheless, estimating the most appropriate empirical numbers has been difficult, in part because there are so many variations in the interpretation of interaction strength (e.g., Yodzis 1988, Paine 1992).

Confusion on the concept of interaction strength can be seen even in its earliest treatments. For example, MacArthur (1972) stated that a strong interactor is a species whose "removal would produce a dramatic effect." This definition is in accordance with what many empiricists attempt to measure (e.g., Paine 1966, 1980, 1992, Power et al. 1985, Morin et al. 1988, Marquis and Whelan 1994, Menge et al. 1994, Wootton 1994b, Laska 1995, Morin 1995, Power et al. 1996), although the relative importance of "strong" vs. "weak" interactors in food webs is somewhat controversial (Menge and Sutherland 1987, Polis 1991, Paine 1992, Goldwasser and Roughgarden 1993, Mills et al. 1993). Yet in his next paragraph, MacArthur (1972) switches his definition, stating that "the competition coefficient, [Alpha], measures how strong the interactions [between two species] are." Although presented side by side, these two ideas of strong interactors are not the same. One represents per capita direct effects of a species, whereas the other describes all of the direct and indirect effects of an entire population. At the same time, May (1972, 1973) tied community stability to species diversity, number of interactions within a community, and interaction strength, the latter defined in yet another way, i.e., the per capita direct effect of one species on the entire population of another species near equilibrium.

To facilitate progress in community ecology, empirical measures of the strength of species interactions need to be incorporated into theoretical models of communities and food webs. For this goal to be attained, there must first be a clear understanding of how different types of empirical data relate to various theoretically relevant concepts. In this paper, we clarify alternative definitions of interaction strength, illustrating their differences and relationships with a simple predator-prey model. We then describe various methods by which empiricists have attempted to measure interaction strength. Finally, we examine the strengths and weaknesses of these approaches, applying them to simulated data similar to the sorts of data empiricists might collect, and investigating how well they recapture the actual interaction strengths underlying the simulations.

CONCEPTIONS OF INTERACTION STRENGTH

The various measures of interaction strength can be placed into four theoretical categories: elements in the community matrix (Levins 1968, MacArthur 1972), elements in the Jacobian matrix (May 1973), elements in the inverse Jacobian matrix (Levine 1976, Yodzis 1988), and elements in the "removal" matrix (following MacArthur 1972 and Paine 1992). In the following section we outline these theoretical measures, show how they are derived, and examine relationships between the four measures, using a two-level food chain model to compare them.

Parameters in a model (e.g., community matrix)

The first measure of interaction strength refers to parameters in dynamic models of species interactions. In simple terms, this concept of interaction strength describes the average direct effect that a single individual of one species has on a single individual of another species (i.e., a per capita effect). MacArthur's idea of [Alpha] as a measure of interaction strength can be extended to a multispecies community using Levins' (1968) derivation of the community matrix, which is the coefficient matrix derived from the generalized form of the Lotka-Volterra equations. For example, consider the following Lotka-Volterra predator-prey functions generalized for consumers (H, herbivores) and their prey (V, vegetation). Population growth of the vegetation is described by