DNkdt f Nk Mk N1 N2 Nf

where Nik is the size of species i population in the kth habitat, functions f (N) represent, as before, a model of local community dynamics, and function Mk(N/, Nf,..., Nh) describes the inflow of migrants from other habitats and the outflow of inhabitants. These functions may well be restricted to the linear ones of the donor components as far as stability is to be studied via the community matrix, that is, by the linearization technique. Thus,

where mkl > 0 is the rate of migration from habitat l to k, mak is the rate of emigration from k to all other habitats, and mi > 0 denotes a species-specific factor of migration intensity that does not depend on the migration route. Being closed w.r.t. migration implies that h mak = mu, k = 1,..., h [25]

l=k or, if the pattern of migration is represented by the h x h matrix Mh = [mkl] with diagonal entries mk = mk [25], the matrix has all column sums zero. System [23] reduces thereafter to h dNk/dt = f (N^ + m.^muN, i= 1,..., n; k = 1,..., h I=1

where the first term governs dynamics in the local habitat and the second one gives the net balance ofmigration (mi = 0 would mean that species i takes no part in migration).

When the environment is symmetric with respect to migration, that is, mkl = mlk for any k = l, matrix M is symmetric, too, and the steady-state equations for system [26] admit a solution

which preserves all the equilibrium population sizes N* of isolated habitats. Whether it preserves their stability properties is a matter of further investigation leading to the metacommunity matrix J(M*).

Linearizing [26] at Ni* results in the nh x nh matrix

of the block-diagonal pattern, where Ih is the h x h identity matrix, symbol ® means the Kronecker product of matrices, and D = diag{mb ..., mn}. The mathematics behind matrix [28] reveals that its spectrum (i.e., the set of all eigenvalues) contains as a subset the spectrum of matrix J(N*). It means that if N* was unstable in the isolated habitat (i.e., if there were eigenvalues of J(N*) with Re A(J) > 0), then the instability is also preserved in system [26]. On the contrary, if matrix J(N*) is stable (all Re A(J) < 0), then for the metacommunity matrix to be stable it is still required that all the remaining Re A(J) be negative too.

So, a symmetric migration pattern cannot provide for greater stability than those in the isolated local habitats. The implications may be twofold: either any stabilizing effects of more complex spatial organization should be searched for in nonsymmetric patterns, or the metacommunity matrix approach is inadequate to the ecological paradigm of persistent metapopulations. In fact, both are true, and the latter deserves further comments.

The paradigm suggests that local populations, though going to eventual extinct, are, in the different habitats, at the respectively different phases of the colonization-extinction process, thus providing for a great enough population size at least somewhere in the heterogeneous environment. In contrast, the metacommunity matrix [28] assumes simultaneous existence of local equilibria in all the habitats, that is, a kind of homogeneous dynamics in the heterogeneous environment. Another kind of mathematics, for example, cellular automata, is therefore needed to model persistent metapopulations. The space dimension, as an additional one in the stability-versus-complexity speculations, does imply greater complexity, which does, in this case, beget stability (in the form of global persistence).

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