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.
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 2 -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 Lotka-Volterra 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 3 . The following four equations show these models and relations between them.
The exponential growth function dP/dt = rP (I.1)
The logistic equation dP/dt = rP (1 -P /K) (I.2)
The two-species LV competition model dP 1 /dt = r 1 P 1 (1 -P 1 /K 1 -α 12 P 2 /K 1 ) dP 2 /dt = r 2 P 2 (1 -P 2 /K 2 -α 21 P 1 /K 2 ) (I.3) 4) The system of LV competition equations describing n-species interactions dP i /dt = r i P i (1 -α i1 P 1 /K i -… -P i /K i -… -α in P n /K i ) (i = 1, …, n) (I.4)
Here P i is the density of ith population individuals (or mass); r i -the specific growth rate parameter. α ik , an interspecific competition coefficient, shows the relative impact of one P k individual on the growth rate of P i population. All intraspecific competition coefficients, α kk , are equal to 1. K i is called the carrying capacity. It is the maximum number of individuals that a given environment may sustain. K i for every species P i in (I.2) and (I.3) is equal to K in the logistic equation for the same species, i.e when it grows without competitors. All these four equations may be considered as the logistic type equations. The exponential growth model is the logistic equation with K → ∞.
Let us begin with a system of n competing species, which evolve according to the LV model (I.4)
This system reaches equilibrium when the following relations hold
On the other hand, it follows from the LV model analysis that while P 1 /K 1 + … + P i /K i + … + P n /K n < 1 (I.6) competing species cannot attain a stable state. Thus at equilibrium P 1 /K 1 + … + P i /K i + … + P n /K n ≥ 1 (I.7) and 1 -P 1 /K 1 -… -P i /K i -… -P n /K n ≤ 0 (I.8)
From here we may write 1 -P i /K i ≤ P k /K k + …+ P i -1 /K i -1 + P i + 1 /K i + 1 + … + P n /K n (i = 1, …, n) (I.9)
Rewrite equations (I.5) in the following manner 1 -P i /K i -α ik P 1 /K i -… -α in P n /K i = 0 (i = 1, …, n) (I.10)
Replacing 1 -P i /K i in relation (I.10) with the right side expression of equation (I.9) we obtain P 1 /K 1 + …+ P i -1 /K i -1 + P i + 1 /K i + 1 + … + P n /K n –α i1 P 1 /K i -…-α i, i -1 P i -1 /K i -α i, i
As (α kk K k -K k ) = 0 for all k we may write instead of (I.11)
Now we transform the equations (I.4) in such a way that the (I.12) expressions would be included into them
or more briefly
The behavior of system (I.4) is quite complex (Strobeck, 1973). Of course n species do not exclude each other if
It follows from (I.14) that
So if we want to be assured that n competitors will coexist permanently it is enough (Strobeck, 1973) to take all competition coefficients such that
I consider relations (I.16) as another way of presenting CEP. This may be true if relations (I.16) is derived from the two-species LV competition model (I.3) -they are the necessary conditions for the coexistence of two species.
Thus mathematics confirms that two competing species P 1 and P 2 according to the LV model may coexist permanently only if intraspecific competition is more intense than interspecific competitio
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