Projection matrices in variable environments: λ1 in theory and practice

Perron–Frobenius theorem for nonnegative matrices, a mathematical foundation of matrix population models, applies when the projection matrix is not decomposable (or equivalently, when it is irreducible), the application yielding the dominant eigenvalue λ1>0 as a measure of the growth potential that a population with given demography possesses in a given environment. In practice, however, the projection matrix often appears to be decomposable (reducible); to calculate λ1 in this case, a principal submatrix should rather be used that corresponds to the reproductive core of the life cycle graph. I call it the reproductive submatrix and demonstrate that, when the reproductive submatrix does not coincide with the projection matrix and if this discrepancy is neglected in a case study, the resulting λ1 may happen to be overestimated. Averaging over a number of annual projection matrices eliminates the false growth rate but raises the problem of choice among the modes of averaging in the estimation of the stochastic growth rate in a stochastic environment. Computer simulation gives a method that avoids the both kinds of problem.