Do "Higher Order Interactions" in Competition Systems Really Exist?

In this paper I reinterpret he experimental results of the widely cited paper by Neill (1974), and show that these results are consistent with a community matrix determined by linear Lotka-Volterra competition equations. I therefore argue that no documented empirical results are presently available which support the concept of "higher order interactions" or multispecies competitive coalitions. The usefulness of mathematical models has often been a point of conflict between theoretical and field ecologists (Gilbert et al. 1976). The perceived complexity of biological interactions can instill a profound distrust for highly simplified mathematical equations manipulated to make predictions about community dynamics and structure. Complex equations, i.e., systems models, may be perceived as more acceptable since they can be as intractable as the living systems they purport to model. However correct this intuition may be, it violates a cornerstone of the scientific method, the "law of parsimony" or Occam's razor. Occam's razor demands that if two hypotheses explain a phenomenon equally well, the simpler explanation will be accepted until it is clearly demonstrated to be less satisfactory than the alternative, more complex explanation. The case in point is the experimental attempt to demonstrate that the simple, linear Lotka-Volterra competition equations are insufficient to "explain" multiple species competition systems. Neill's position, with which I agree, is that if the Lotka-Volterra model is inadequate for a laboratory system it is necessarily inadequate for field systems. His experimental system was well designed and is adequate to demonstrate higher order interactions. However, Neill's statement that "the burden of proof for this contention (that higher order terms are unimportant . .) rests with those arguing the Lotka-Volterra equations are sufficient. " (p. 401) is not acceptable because it violates Occam's razor. It accepts the more complicated explanation before the simpler one has been demonstrated to be unsatisfactory. If data are collected that indicate nonlinear interactions in a competition system, there are at least two categories of nonlinear modifications in the LotkaVolterra equations which can be invoked to explain the results. The first, and less serious class of modification would require nonlinearities in the intraspecific growth dynamics of some or all of the competing species (less serious because the effect of all other species remains a linear function of their densities). A simple modification might be the addition of an exponential parameter in the intraspecific portion of the growth equation (Pella and Tomlinson 1969; Gilpin and Ayala 1973; Ricklefs 1973; Ayala et al. 1973; Gilpin et al. 1976; Thomas et al. 1980). The second class of modification is "higher order interactions" where additional competition parameters are added to the equations. These parameters depend on organisms having different competitive effects when acting in coalitions and when acting alone (e.g., Vandermeer 1969; Ayala et al. 1973). Should such effects be found, the body of theory and prediction developed from the community or