Lotka Volterra Population Equations and Dissipative Matrices

The term 'Lotka-Volterra population equations' is associated nowadays with the following particular case of system [1]:

where parameter e, denotes the intrinsic rate of population natural increase for species i and —7, indicates the effect species j renders upon the per capita growth rate, dN/dt/ N, of species i (the minus sign is introduced here to save the original notation of V. Volterra). The parameters of [9] constitute n vector E = [e1,..., en]T and n x n matrix -r = [—7 j ] called the interaction matrix. A feasible equilibrium N* exists if there is a positive solution to the system of linear algebraic equations (LAEs)

written here in the vector-matrix form. Calculated at N*, the community matrix [4] equals n f

having the sign pattern in common with the interaction matrix — r. When matrix J is stable, N* is locally Lyapunov stable too.

V. Volterra studied a restricted class of systems [9] which he defined via a special property of the community matrix: matrix J is called 'dissipative' (or 'diagonally stable' in some texts) if there exists a diagonal matrix B = diag{3i,.. ., 3n} with positive entries 3i such that matrix BJ generates a negative definite quadratic form (BJx, x), or equivalently, matrix (BJ + JTB)/2, the 'sym-metrizor' of BJ, is negative definite. This property guarantees that function is a Lyapunov function for the equilibrium point N* and any positive vector N. It means that equilibrium N* is stable globally for any large perturbation of N*. The existence of N* thus implies its global stability, signifying globally stable coexistence of n species.

If, however, some of the components happen to be nonpositive in the solution of system [10], then there still exists a maximal subset of k < n species which yields a positive solution N' of the system reduced to the corresponding principal submatrix r' and 'subvector' E'. Since any principal submatrix of a dissipative matrix is itself dissipative, the subset of k species proves to be globally stable, hence invulnerable to invasion by any other of the n — k species. Moreover, given a dissipative interaction matrix r and individual species characteristics e, a proper theorem guarantees that a stable k-species composition is unique.

Since the set of dissipative matrices is closed w.r.t. multiplication by any diagonal matrix D with positive entries, the community matrix and the interaction one do or do not belong to the dissipative set simultaneously. This enables tackling the interaction matrix r before calculating the Jacobian matrix [11]. For instance, if a 2 x 2 matrix of type [7] is stable, then it is dissipative too. A criterion that provides for a 3 x 3 matrix to be dissipative is more sophisticated, so that mere stability is no longer sufficient for the matrix to be dissipative. There are some necessary conditions and some sufficient ones for a matrix to be dissipative in the general case. For example, a matrix has no chance to be dissipative if it has a non-negative diagonal entry; in other words, only self-limited species can form a dissipative community. But if,on the other hand, the negative diagonal entries are so great in magnitude that each one exceeds the sum of off-diagonal entries (in modulus) along its row, then the matrix is dissipative. In other words, self-limitation in each species stronger than all interspecies effects on that species (the so-called 'diagonal dominance' in the matrix) is a sufficient condition for the community to be dissipative.

Now, if the diagonal dominance, rather than mere stability, were analyzed in the randomly constructed webs of the previous section, then greater values ofthe complexity measure s2nC would apparently decrease the share of matrices in which the (unitary) diagonal is still dominating. So, greater complexity does counteract the global stability, too.

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