Does environmental variability limit niche overlap ? ( limiting similarity / invasion / stochastic competition models / random environments / stochastic approximations )

A stochastic theory of limiting similarity is presented that attempts to quantify the relationship between tolerable niche overlap among competing species and the degree of environmental fluctuation. The theory is based on a heuristic analytical approximation that provides conditions under which a rare invadin species can increase in the presence of a community of established competitors. The major qualitative conclusion, derived from investigating two symmetric, discretetime, stochastic analogs of the Lotka-Volterra competition equations, is that weak to moderate stochastic variation does not appear to limit significantly the similarity of competing species. This result is in sharp contrast to the conclusions of May and MacArthur's pioneering study of stochastic limiting similarity. A possible reason for this discrepancy is explored. Ecologists have long been interested in relating the number of competing species that coexist in a given region to characteristics of its biotic and abiotic environment. Among the factors suggested are: (i) the diversity and relative abundance of resource types available; (ii) the tolerable niche breadth (i.e., the extent to which species can be specialists rather than generalists); and (iii) the tolerable degree of niche overlap. The possible importance of this last factor in accounting for tropical species diversity was stressed by Klopfer and MacArthur (1). After presenting morphological data suggesting that sympatric bird species are more similar in the tropics than in the temperate zone, they argued that an increased capacity for niche overlap may arise from the greater constancy of tropical environments compared to temperate. The stochastic theory of limiting similarity presented below is an attempt at quantifying the relationship between tolerable niche overlap and the level of environmental fluctuation. Its methods and conclusions differ substantially from those of previous studies. Following previous usage (1-4), "degree of niche overlap" will be used as a synonym for intensity of interspecific competition. However, no particular mathematical relationship between resource utilization and the degree of competition is assumed, and the analysis is based directly on competition coefficients. The first mathematical investigation of Klopfer and MacArthur's conjecture was carried out by May and MacArthur (2) and subsequently elaborated by May (3, 4). They began by inserting Gaussian white noise into a reparameterized version of the standard Lotka-Volterra competition equations dXi/dt = riXi1_E >ai1X1/Kt) i = 1,2,. n. [1] Here Xi denotes the abundance of species i, r1 is its growth rate, Ki is its carrying capacity, and atij measures the degree of competition (niche overlap) between species i and j. By applying a qualitative stochastic stability criterion proposed in ref. 5, they concluded that environmental variability does limit the niche overlap of competitors. Moreover, they asserted that, if many species compete along a one-dimensional resource axis and competition is related to niche overlap via a "micromodel" yielding aij a('iJ)2, then a significant limit on niche overlap is imposed by even a very small degree of stochasticity and the limit is relatively insensitive to the degree of variation as long as it is not too extreme. The validity of these conclusions is questionable for various reasons. Feldman and Roughgarden (6) criticized the reparameterization of Eq. 1 used. They also showed that, for single species models, results qualitatively different from those in ref. 5 could be obtained by applying May's stationary distribution stability criterion to an alternative model based directly on Eq. 1 or by reinterpreting May's random logistic according to another mathematical convention. Problems arising from the ambiguity of stochastic models heuristically obtained by adding white noise to ordinary differential equations were discussed at length in ref. 7. There it was argued that the interpretation used by May and MacArthur can be justified by viewing the heuristically derived models as approximations for biologically more accurate stochastic difference equations. However, to circumvent the difficulties associated with stochastic differential equations, the present study is based on stochastic versions of the following discrete-time analog of Eq. 1: Xit+ 1Xit = riXi,t (1-E aijXj,t/Ki) i= 1,2,... n.