## Niche Overlap and Stability of Horizontally Structured Communities

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

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 , 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 , N* is locally stable if the following matrix is stable:

But matrix —A with elements  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 .

Matrix A defines the linear transformation of the vector space, which is also denoted by A. Geometrically, for system  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 Figure 5 Feasible equilibrium N* does exist as a positive solution of system  for n = 3.

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  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)  becomes the measure of community stability, and it does decrease when the complexity parameters s, n, C increase. The complexity would again counteract stability. 