While trophic chains, with their ascending transfers of energy, can be naturally viewed as 'vertically' structured communities, the interactions among species within a single trophic level represent a 'horizontal' structure. Typical example is a community of several species competing for vital resources (space, light, nutrients, etc.) as far as their respective niches overlap in the space, R, of those resources.

Let a function K(c) indicate a level of the (multidimensional, in general) resource available at point c of the space. If, in formal terms, species i (i = 1, 2,..., n) has a preferential value mi of its resource, while the consumption at other points c is distributed around m^ as the mean value in accordance with a unimodal function fi (c), called a 'utilization function' (see Figure 4a), then a species i individual

Figure 4 One-dimensional illustration of the resource space and utilization functions for two species: (a) specialist species (1) and generalist species (2); (b) two close specialist species. The shaded area indicates niche overlap.

encounters a species j individual in competition for a small portion dc of the resource at point c with probability f(c) x f(c)dc. Magnitude of the negative effect that species j population renders upon the per capita growth rate of species i population is hereafter expressed as the integral of the local competition over the whole resource space,

System [9] of Lotka-Volterra population equations then takes on the form m/d/ = r,N,(^ 1 - aVNj/K j , i = 1,..., n [17]

where ri > 0 is the intrinsic rate of natural increase in species i population,

is the carrying capacity of the environment for that species, or the niche capacity, and A = [aj] is the matrix of competition coefficients [16], symmetric by construction.

Feasible equilibrium N* > 0 is a positive solution to the system of LAEs:

with vector K = [Kb ..., Kn] > 0 on the right-hand side. As was noted in [11], N* is locally stable if the following matrix is stable:

But matrix —A with elements [16] has an even stronger property: it is dissipative in the sense described in the section entitled 'Lotka-Volterra population equations and dissipative matrices'. If, indeed, x = [x1,..., x„]T = 0 is a arbitrary n-vector and B = I, then, due to the additive property of integration, the quadratic form n n f

is always positive. Therefore, equilibrium N* > 0 is globally stable once it does exist, and it is the latter event that predetermines stable coexistence of n species in the competition community [17].

Matrix A defines the linear transformation of the vector space, which is also denoted by A. Geometrically, for system [18] to have a positive solution, it is necessary and sufficient that vector K belongs to the interior of the n-hedral cone to which the transformation A compresses the cone of all possible model states, that is, the positive orthant R+ n. The wider the cone AR+ n, the more likely that vector K

gets into the cone. Figure 5 suggests an illustration for n = 3 and cross-section of both cones by a plain simplex £3 = {N + N2 + N3 = const}. The 3-hedral cone AR+ 3 cuts a triangle, 03(A), out of the simplex, and a measure of the cone 'width', ^(A), is apparently given by the ratio of the respective areas, which can be shown to equal

M(A) = 03(A)/£3 = det(A)/[||A(1)||s||A(2)||s||A(3)||s]

where ||A(/)||£ denotes the sum of column j elements (j = 1,... , n = 3, the expansion to the general n being straightforward). It measures the variety of species-environment parameters Ki for which a given pattern of resource utilization, called 'species packing' in the resource space, provides for stable coexistence of the n competitors.

By definition, 0 < ^(A) < 1, and the maximal value yu(A) = 1 is attained when A = diag{an ,..., ann}, that is, when n self-limited species do not overlap in their niches. On the other hand, if the overlap is significant and the utilization functions are alike (Figure 4b), then a j is fairly close to the self-competition coefficients an, aj. The determinant of A is close to zero in this case, while the cone AR+ n becomes fairly thin, concentrating around the central direction, that is, vector [1,..., 1], thus decreasing the diversity of stable vectors K. Algebraically, for example, matrix [7] results in

if a and b tend to 1, then ^ tends to zero. The denser the species packing and the more similar the utilization functions, the closer is ^ to zero.

If now the species packing were constructed randomly, in the sense described in the section entitled 'Population equations, community matrix, and Lyapunov stability', then the resulting matrix A would always be stable (even dissipative), but the chance of community stability would vary with vector K. Correspondingly, ^(A) [21] becomes the measure of community stability, and it does decrease when the complexity parameters s, n, C increase. The complexity would again counteract stability.

Was this article helpful?

You Might Start Missing Your Termites After Kickin'em Out. After All, They Have Been Your Roommates For Quite A While. Enraged With How The Termites Have Eaten Up Your Antique Furniture? Can't Wait To Have Them Exterminated Completely From The Face Of The Earth? Fret Not. We Will Tell You How To Get Rid Of Them From Your House At Least. If Not From The Face The Earth.

## Post a comment