The logistic equation and a critique of the theory of natural selection

Species coexistence is one of the central themes in modern ecology. Coexistence is a prerequisite of biological diversity. However, the question arises how biodiversity can be reconciled with the statement of competition theory, which asserts that competing species cannot coexist. To solve this problem natural selection theory is rejected because it contradicts kinetic models of interacting populations. Biological evolution is presented as a process equivalent to a chemical reaction. The main point is that interactions occur between self-replicating units. Under these assumptions biodiversity is possible if and only if species are identical with respect to the patterns of energy flow in which individuals are involved. I. The paradigm Competition theory, probably the oldest ecological theory (Gause, 1934b; Lotka), makes a fundamental link between ecology and evolutionary biology. The idea that competing individuals are evolving individuals, proposed by C. Darwin in the form of natural selection theory (Darwin, 1859), makes the background of contemporary evolutionary thought. Now this is widely accepted scientific model that explains the mechanism and describes the consequences of evolution of living things (Calow, 1983; Khare & Shaulsky, 2006). According to Darwin the more similar species are the more severely they compete. This should lead to the extinction of some less adapted intermediate forms and to the further divergence of competing species. I name this statement on the relationship between species similarity and the strength of competitive interactions between them Darwin’s competition principle (DCP). As we will see later it has close connections with other biological rule – the central law of ecology – which has been called the principle of competitive exclusion (CEP) (Gilbert et al., 1952; Hardin, 1960; Schoener, 1982). CEP, also called Gause’s hypothesis, states the following: If we replace some individuals of a particular species, living in a given environment, with individuals of another species then these two populations cannot live together permanently – one species will be excluded, partially or completely, from that environment. In the formulation of classical CEP and its more recent version – the limiting similarity theory (McArthur, 1967; Brown, 1981; Giller, 1984) – a considerable part took the LotkaVolterra type (LV) species competition models (Gause; Lotka, 1925; Volterra, 1926; Pianka, 1978; van der Vaart, 1983). These equations have had a big impact on ecological theory and up to now are widely used (...). The LV models are a natural extension of the logistic equation, which in turn is a modification of an exponential growth formula called the Malthus’ growth law. The following four equations show these models and relations between them. 1) The exponential growth function