## Population Equations Community Matrix and Lyapunov Stability

The state of a community comprising n > 2 species is usually represented as n-dimensional vector N(t) = Ni(t),..., Nn(t)]T, the component N(t) (i = 1, 2,..., n) being the population size of species i at time moment t (superscript T means transposing since a technical tradition is to tackle column vectors). When the dynamics of N(t) in a constant environment is governed (modeled) by a system of n autonomous (i.e., time-invariant) ordinary differential equations (ODEs), dN/dt = f (N) (i = 1,..., n) 

it is supposed to have a feasible equilibrium N* = [N*,.. .,Nn*]T> 0 (i.e., with strictly positive components N* > 0) as a proper solution to the (nonlinear, in general) system of equations f (N*)= 0 (i = 1,..., n)

In the vicinity of N*, where deviations x(t) = N(t) — N* are not great, a nonlinear system  can be approximated by means of a linear one, namely, dxi/dt = [Jx]i (i = 1,

with a matrix J = [aj ] of size n x n (n rows and n columns) calculated as j(N*) = [Qf, /qNj j N •

and called, in mathematics, the Jacobi matrix (of system  at point N*). In theoretical ecology, it is called a community matrix for the following reason.

Any element aij(i=j) of the matrix means, by its construction , a special rate of how the population size of species i changes in response to variation in the species j population size, the other populations not varying. In other words, matrix element aij does indicate whether species i increases, decreases, or remains indifferent in its population size with regard to population growth of species j and how strong this effect is in magnitude. Element aji, standing symmetrically about the main diagonal of the matrix, represents, by the very same definition, a reciprocal effect that species j exerts upon species i. On the main diagonal we have i = j, so that diagonal entries aii represent the effects that populations exert upon themselves, that is, intraspecies regulations by the population size.

These mathematical meanings are quite consonant with the ecological classification of species interaction types based on the signs of the direct and reciprocal effects (Table 1). A great variety of kinds of interaction are thereafter reduced to six different pairs of symbols selected from the three possible ones (+, —, or 0). Given a community matrix, those pairs should be taken from each pair of symmetric entries, thus characterizing the pattern ofspecies interactions in the given community. And, vice versa, given a community as an ordered list of n constituent species, each with a known subset of interacting species within the community, the overall pattern of species interactions can be consequently, row by row top-down, constructed by Table 1 rules applied to each pair ofsymmetric elements to result in an n x n matrix of 3-valued off-diagonal entries. Adding the known signs of species self-regulations to the a priori zero main diagonal completes the construction.

For example, given a hypothetical list of three species -(1) a self-regulated commensal benefiting from both prey species (2) and its predator (3) the community matrix takes on the form

where positive parameters a, b, c,... denote unknown magnitudes of the effects. Given a trophic chain of five species ascending from a primary resource species (1) with self-limitation to the top predator one (5), the community matrix takes on the form

 Sign of the effect Sign of the effect that species 1 that species 2 exerts upon exerts upon species 2 species 1 Terminology — — Competition — + Resource-consumer, prey-predator, host-parasite + + Mutualism, symbiosis — 0 Amensalism + 0 Commensalism 0 0 For a competition community of three species ordered within its niche space, the community matrix, when scaled so that each intraspecies competition coefficient equals 1, looks like The community matrix plays the key role in 'complexity-versus-stability' studies since; in addition to representing the community structure, it also gives a key to analyze stability in model . When the initial state N(0) coincides with a feasible equilibrium point N*, it remains there (unless perturbed for an external reason) as the rates of changes are all zero at that point. But if the initial state is distinct (perturbed) from N*, further dynamics of N(t) may be quite variable. It can be interpreted as stable coexistence of n species if neither N(t) vanishes, nor increases indefinitely. If, moreover, those perturbed initial states N(0) which are close enough to the unperturbed state N* return eventually (i.e., when t tends to infinity) into a small vicinity of N*, either mono-tonically or in oscillatory mode, then N* is said to be Lyapunov stable. Otherwise, if any small perturbation of the initial state does increase with time, either monotoni-cally or in oscillatory mode, then the state is called unstable. This is the simplest and the most tractable notion of stability/instability in mathematics, and this kind of stability serves as an immediate formal analog for stable species coexistence under local perturbations of the community state. A key theorem of stability analysis states that equilibrium N* is Lyapunov stable in system  if matrix J  is stable, that is, all its eigenvalues A have negative real parts, and it is unstable if there is an eigenvalue with a positive real part. (The eigenvalues of an n x n matrix J are n complex numbers with special properties, which can be found as n roots of the so-called characteristic polynomial for matrix J, i.e., as n solutions to the polynomial equation det[AI - J] = 0, where det means the matrix determinant and I denotes the identity matrix, the diagonal n x n matrix whose diagonal entries are all equal to 1. Standard routines of computer software can calculate the eigenvalues in most practical cases.) Matrix J1 , for instance, is not stable as its three eigenvalues are A1j2j3 = - a, ±i V^d, representing a boundary case with its Re A23 = 0, where equilibrium N* may not be stable. Matrix J2  is stable for any magnitudes of its nonzero entries, that is, for any values of positive parameters a, b, c, ... (yet the algebra to prove this is not trivial). All the eigenvalues of matrix J3  are real numbers, and they are all negative if b <1 and a < \J(1 + b)/2. Correspondingly, the feasible equilibria in model communities with stable matrices J2 or J3 are locally stable. So, the community matrix is both a certificate of interaction structure in a multispecies community and a tool to analyze the community stability. It bears comparable parameters of complexity and certain quantitative indices (eigenvalues) of stability. This duality enables formal speculations on the stability-versus-complexity theme in the following way. Given a number n of species in a community, the 'universe' of all topologically possible community patterns represents the linear space of n x n matrices with real elements (zero or nonzero, positive or negative). If a matrix, J, is stable, then so is the matrix pJ for any positive number p. It means that the fraction of stable matrices in the 'universe' is the same as that fraction in a compact 'subspace' of the ''universe''. How does it change as a function of complexity parameters? A certain answer was given in the framework of 'randomly constructed webs'. When trying to see whether interactions among species tend to stabilize or destabilize a community, it is logical to assume each species population to be stable in isolation, that is, each species to be self-limited, hence each diagonal entry of the community matrix to be negative. All of them can be put equal to -1 without loss of generality, thereby scaling each row of the matrix w.r.t. the diagonal entry. The fraction, C, of nonzero offdiagonal elements to the number of all ones, n(n — 1), is called connectance. Among the nonzero entries, the average value s of their magnitudes (absolute values) is the mean strength of interactions. A community matrix, as the synonym to a random web with fixed C and s parameters of complexity, can thereafter be constructed around the fixed diagonal, by choosing at random, first, whether an off-diagonal entry is nonzero (with probability C) and, second, choosing the nonzero value from a random distribution around zero with the variance equal to s2. Once a web Numerical matrix is constructed, its eigenvalues are calculated, hence stability/instability is certified. The probability, P(n, C, s), that a random web is stable can therefore be estimated as the share of stable matrices in the outcome of the procedure. For great enough n, it was proved to have a jump-wise pattern, from 0 to 1, at a threshold value of sVwC": In other words, the magnitude of s2C, with its apparent relation to complexity, serves as an inverse measure of stability: for a fixed n, it suggests the same level of stability likelihood for communities of many weak interactions (greater C, smaller s) as for those of a few strong interactions (smaller C, greater s); when the complexity becomes great enough to provide for s2nC> 1, it makes the stability improbable. Whether the locally stable systems can resist nonlocal (i.e., large) perturbations is an issue of global stability analysis, relied mostly on searching for a function of N with special properties, called a Lyapunov function. In general, there is unfortunately no regular method for searching that would be as universal as the eigenvalue calculations. However, in some kinds of community models , a locally stable state appears to be globally stable, and the next section considers an important class of such models defined via the community matrix. 