13. Potential of Branching Processes as a Modeling Tool
225
Then, in the case of the subcritical Bienaym´ e-Galton-Watson BP, if we skip the
technicalities of eigenvalues and vectors of infinite matrices, (1, 0, 0, . . . ) is an
eigenvector of the matrix P, associated with the dominant eigenvalue 1, and the
quasi-stationary distribution (b k ) k∈N * is an eigenvector of Q associated with the
dominant eigenvalue of Q, m. The same holds, for example, for density-dependent
BP, under further conditions, by replacing m by the asymptotic growth rate λ.
225
Then, in the case of the subcritical Bienaym´ e-Galton-Watson BP, if we skip the
technicalities of eigenvalues and vectors of infinite matrices, (1, 0, 0, . . . ) is an
eigenvector of the matrix P, associated with the dominant eigenvalue 1, and the
quasi-stationary distribution (b k ) k∈N * is an eigenvector of Q associated with the
dominant eigenvalue of Q, m. The same holds, for example, for density-dependent
BP, under further conditions, by replacing m by the asymptotic growth rate λ.