## Qualitative Stability and Sign Stable Community Matrices

Another stronger-than-Lyapunov notion of matrix stability arises from the idea ofstability which is caused by the qualitative pattern of species interactions alone and does not depend upon the strength of those interactions. In matrix terms, it means that stability is a function of the matrix sign pattern, Sign J, and does not depend (unlike eigenvalue calculations in particular cases) upon quantitative magnitudes of matrix elements. The formal definition defines a matrix as sign-stable if it is (Lyapunov) stable together with any other matrix B which is sign-equivalent to J. Given a fixed sign pattern, Sign J, the sign-equivalent matrices constitute a family of infinite number of matrices which differ from one another only in magnitudes of their nonzero entries.

For example, matrices of the form  and , with any positive values of parameters a, b, c,..., do represent such families with the sign patterns

Sign J1

=

0

+ 0

+ "

0

+

0

-

-

0

+

0

-

Sign J2

=

0

+

0

0

0

+

0

0

0

respectively, but

matrices



+ i do not (because of the imposed symmetry in their off-diagonal entries). As was noted above, neither matrix of the kind  is stable, but any matrix  is stable indeed, thus being sign-stable by definition.

The sign pattern of a matrix can also be represented as a signed directed graph (SDG) with numbered vertices and signed directed links. Figure 1a shows the SDG of any matrix  and Figure 1b does so for . An SDG represents the whole family of sign-equivalent matrices and the family does or does not possess the sign stability. Sign stability thereafter becomes a property of the SDG, and it must therefore be verifiable in terms of the SDG.

For n = 2, there are only two sign patterns (up to re-numeration of the spp.) which are sign-stable:

For n > 2, the following criterion (i.e., a necessary and sufficient condition) for sign stability was established in the late 1970s and 1980s. An indecomposable matrix J (i.e., a strongly connected SDG) is sign-stable if and only if it meets each of the five terms below:

1. there is no one '+' sign and at least one '—' sign on the main diagonal;

2. for each pair of symmetric entries their product is either zero or negative;

3. in the SDG, there are no directed loops that would comprise three or more vertices;

 S1 = - - ; S2 - - .+ 0 Figure 1 SDGs for particular matrices: (a)  and (b) .

4. the SDG admits decomposition into several disjoint 1- or 2-vertex loops;

5. when considered as a nondirected graph, the SDG fails the Jeffries' color test explained below.

Jeffries' color test. A connected nondirected graph passes the color test if all its vertices can be colored black or white to obey the following three rules:

(a) all self-looped vertices are black;

(b) there exist white vertices, each one being linked to another white vertex; and

(c) if a black vertex is linked to a white one, then it is also linked to at least one other white vertex.

Figure 2 suggests schematic images of these rules, and Figure 3 illustrates the test in some particular cases.

When a matrix is decomposable, it should be decomposed into the diagonal blocks corresponding to strongly connected components of the original SDG and the criterion should be verified in each of the blocks. For example, matrix  decomposes into [—a], which is trivi-" 0 - d'

ally sign-stable, and

which violates rules (1)

and (5), thus lacking sign stability (and even mere stability because of pure imaginary eigenvalues). Matrix  is

Figure 2 Schematic images of the rules of Jeffries' color test.

Figure 3 Application of the color test in particular cases: (a) matrix  and (b) matrix .

indecomposable and conditions (1)-(4) hold apparently true for its SDG; the obligatory black color of vertex 1 breaks rule (c) of the color test (Figure 3 a), hence the matrix is sign-stable.

Matrices of pattern  for n = 2, 3,... appear in a mathematical theory of trophic chains, that is, systems where energy is successively transferred as consumed food from the first level of primary producers up to the top predator at the nth level. The theory, which relies upon a more sophisticated system of population equations than mere Lotka-Volterra ones, predicts stable existence of a q-level trophic chain (q = 2,..., n) as a function of resource input rate at the lowest level: greater rates sustain the greater number q < n of trophic levels, the theory thus conforming reality. In reality however, the maximum number n can hardly exceed n = 5-6 because of progressive losses of the energy transferred to the next trophic level. Practical estimates of chain parameters suggest the existence of a q-level equilibrium in the model with certain ranges of variability in the parameters, while the criterion of sign stability guarantees the equilibrium to be stable within the ranges.

If, however, the sole self-limitation loop in the SDG for matrix  were at vertex 3, as it is for matrix

then the graph would pass the color test (Figure 3b); hence, the criterion would fail and the matrix would not be sign-stable. Numeric matrix  is just a specimen from the sign-equivalent family which is not stable as it has A = ±i among its eigenvalues.

To comment on the general criterion ofsign stability in terms of complexity parameters n, C, s, note first that parameter s is irrelevant for sign stability by definition. But, second, as prompted by Figure 3, it is only for great enough n (starting from n = 5) that the color test for an SDG with a self-limited species may be positive; hence, the criterion may fail. Third, the more links there are in the SDG (the greater C), the greater is the chance for three or more vertices to occur in a directed loop and hence for the criterion to fail again. So, greater complexity does counteract the qualitative stability also. 