Indirect Mutualism: Variations on a Theme by Stephen Levine

Although some of the most spectacular interspecific interactions in nature are obviously mutualistic, relatively little research, empirical or theoretical, has been aimed at understanding this basic and perhaps prevalent form of interaction (Risch and Boucher 1976). In this paper I attempt o extend the work of Levine (1976) in the context of the basic summary provided by Vandermeer and Boucher (1978), thereby providing a theoretical framework for the concept of "indirect mutualism" (S. James and D. Boucher, in prep.) or "niche enhancement" (Dodson 1970). I furthermore suggest that this form of interaction is likely to be extremely important in community organization, at least at some trophic levels. Vandermeer and Boucher (1978), using simple linear analysis, have categorized the basic forms of mutualistic interactions. In a fashion similar to Gause and Witt's analysis of competition (Gause and Witt 1935), Vandermeer and Boucher presented eight qualitatively distinct outcomes, predicted by a simple two-species system of differential equations. Their eight cases are summarized in table 1. Two features of their esults are of particular importance here, first hat the interactions may be stable or unstable, and second that mutualists may be facultatively mutual or obligately mutual. Note that the local stability of the system (the row headings as they appear in the first col. of table 1) has very little to do with the biological outcome in the usual sense. That is, in the case of interspecific competition, ifa local analysis indicates tability coexistence of the two species is automatically insured. However, no such correspondence exists with mutualism, as can be seen in table 1. Biological coexistence is possible when the system is mathematically unstable. That mutualists can be either obligate or facultative isobvious, and theoretically crucial since biological outcomes depend on this dicotomy (see table 1). The phenomenon of obligate mutualism has been modeled by assuming a negative carrying capacity (Vandermeer and Boucher 1978), an approach continued here. Levine (1976) extended the popular consumer esource quations of MacArthur (1968, 1972) to include interactions between resources. Levine's equations are shown diagramatically infigure 1. Levine noted that for a wide variety of parameter values the two consumers are "effectively" mutualistically associated with one another, as long as the resources are competitively associated. This effectively positive relationship shere called indirect mutualism, and is generated in an intuitively obvious fashion. If a resource required by a consumer is maintained at a relatively low biomass through competition with another resource, any factor