Habitat destruction, habitat restoration and eigenvector-eigenvalue relations

According to metapopulation theory, the capacity of a habitat patch network to support the persistence of a species is measured by the metapopulation capacity of the patch network. Mathematically, metapopulation capacity is given by the leading eigenvalue lambda(M) of an appropriately constructed non-negative n x n matrix M, where n is the number of habitat patches. Both habitat destruction (in the sense of destruction of entire patches) and habitat deterioration (in the sense of partial destruction of patches) lower the metapopulation capacity of the patch network. The effect of gradual habitat deterioration is given by the derivative of lambda(M) with respect to patch attributes and may be straightforwardly evaluated by sensitivity analysis. In contrast, destruction of entire patches leads to a rank modification of matrix M, the effect of which on lambda(M) may be derived from eigenvector-eigenvalue relations. Eigenvector-eigenvalue relations have previously been analyzed only for symmetric matrices, which restricts their use in biological applications. In this paper I generalize some of the previous results by deriving eigenvector-eigenvalue relations for general non-symmetric matrices. In addition to the exact eigenvector-eigenvalue relations, I also derive eigenvalue perturbation formulae for rank-one modifications. These results lead to simple and intuitive approximation formulae, which may be used e.g. to assess the contribution of particular habitat patches to the metapopulation capacity of the landscape. The mathematical results presented are not restricted to the metapopulation context, but they should find a number of useful applications in biology, engineering and other applied sciences, where the removal (or addition) of matrix rows and columns often corresponds in a natural manner to decreasing (or increasing) the degrees of freedom of the focal system.