The Waterwheel in the Waterfall

A fundamental problem in evolutionary ecology research is to explain how different species coexist in natural ecosystems. This question is directly related with species trophic competition. However, competition theory, based on the classical logistic Lotka-Volterra equations, leads to erroneous conclusions about species coexistence. The reason for this is incorrectly interpreted interspecific interactions, expressed in the form of the competition coefficients. Here I use the logistic Lotka-Volterra type competition equations derived from the so called resource competition models to obtain the necessary conditions for species coexistence. These models show that only species with identical competitive abilities may coexist. Due to such relations between competing species ecosystems biodiversity decreases in the course of evolution. World ecosystems are rich in species. But how is species diversity maintained? I reduce this problem to the following question. When do competing species coexist? Considering the formal description of species trophic competition we encounter with two kinds of interactions. Either populations of n species xi use a common resource or m species of resources Rj are exploited by the same consumer. Traditionally only the first case biologists call competition. The second one was named apparent competition (Holt, 1977). I use the terms divergent competition (d-competition) and convergent competition (c-competition) for them respectively. cd-competition describes a fully connected food web of n consumers and m resources. Suppose for simplicity that the total mass density M = ΣRj + Σxi of the above food web components remains constant. Then cand d-competitive interactions may be formalized by a set of equations dRj/dt = ρRj Σβi xi (1) dxi/dt = βi xi qi xi where ρ, βi , βi and qi are functions such that ΣρRj = Σqi xi and βi = Σβi . Functions βi j (and βi) describe mass transformation due to interactions between Rj and xi. From (1) I derive the relative competition strength functions Φk for both types of interactions Φk = cu + dku, k = 1, ..., p, p ∈ {m, n} (2a) or Φk = Φind + Φdir,k (2b) Here Φk is equal to (φkuk – duk/dt)/φkuk; φk is some function of the growth rate of population mass density. The terms on the right side of the equation (2a) represent the relative strength of the two sides of competition process. I call them indifferent and directed competition, respectively. u is a vector of either consumers xi or resources Rj densities expressed in the same units. c and dk are vectors with the components ck and dkl (l = 1, ..., k). ck, an indicator of indifferent competition, describes the competitive ability of a population density unit of kth species. dkl = ck – cl, an indicator of directed competition, may be interpreted as a competition potential. ck has the following expressions cj -1 = Rj Σqi xi /Σβi xi for c-competitors (3) ci -1 = M qi ΣRj /βi for d-competitors Equations (2) have the same form as the logistic Lotka-Volterra competition models, employed as a mathematical description of Darwinian evolutionary theory (Gause, 1934). However, as they were not related with resources exploitation explicitly (Grower, 1997), their interpretation was incorrect and wrong conclusions have been drawn using these classical competition models. Particularly the errors are related with the description of the nature of interspecific competition coefficients, usually indicated as αkl. As we see from (3) all such coefficients in the model (2) are equal to 1. As a result of this misunderstanding the theory of limiting similarity arose (MacArthur and Levins, 1967).