multitype branching and the Perron-Frobenius mean matrix
/ peh-ROHN FROH-beh-nee-oos /
Real populations are not made of interchangeable individuals: there are males and females, infected and recovered, juveniles and adults, particles of different energies. The multitype branching process generalizes Galton-Watson to d types, where an individual of type i produces a random vector of offspring of all types. The astonishing fact is that the entire subcritical/critical/supercritical trichotomy survives, with the single mean m replaced by the leading eigenvalue of a mean matrix, governed by the Perron-Frobenius theorem.
Let there be d types. An individual of type i has offspring described by a probability law on vectors in Z^d_{>=0}; let M be the d-by-d mean matrix with entries M_(ij) = expected number of type-j children of a type-i parent. Then the vector of expected generation-n populations is the matrix power: E[Z_n given Z_0] = Z_0 M^n (row vectors). If the process is positively regular (some power M^k has all entries strictly positive — every type can eventually beget every type), the Perron-Frobenius theorem gives a simple, positive leading eigenvalue rho (the spectral radius) with strictly positive left and right eigenvectors u and v. This rho plays exactly the role of m: the process is subcritical, critical, or supercritical according as rho < 1, rho = 1, or rho > 1. Extinction is certain when rho <= 1, and possible-survival when rho > 1, with a vector q of type-dependent extinction probabilities solving the system q_i = f_i(q) where f_i is the multivariate offspring generating function of type i. On survival the normalized population vector aligns with the right eigenvector v, and Z_n / rho^n converges to W times v where W is a scalar Kesten-Stigum martingale limit (nondegenerate under a multitype x log x condition).
Why it matters: this is the engine behind matrix population models in ecology (Leslie matrices for age-structured populations), multitype epidemic models with several host classes, the neutron-transport (criticality) calculations for nuclear reactors, and cell-lineage models with distinct cell types. The deep message is that the long-run growth rate is one number, the Perron eigenvalue, and the stable type-composition is its eigenvector. The crucial hypothesis is positive regularity (irreducibility plus aperiodicity of M): without it the leading eigenvalue may not be simple or its eigenvector not strictly positive, the clean type-balance can fail, and the trichotomy must be stated component-class by component-class. Do not apply the single-rho criterion to a reducible mean matrix.
An age-structured population with juveniles (type 1) and adults (type 2): juveniles become adults with probability s, adults produce b juveniles each. The mean matrix M has first row (0, s) and second row (b, 0) (rows = parent type, columns = child type), with leading eigenvalue rho = sqrt(s b). The population grows if s b > 1, is critical at s b = 1, and declines if s b < 1 — a single product of survival and fertility decides the fate.
The Perron-Frobenius eigenvalue rho of the mean matrix M replaces the scalar mean m; rho vs 1 sets criticality.
The single leading-eigenvalue criterion needs positive regularity (M irreducible and aperiodic). For a reducible mean matrix the leading eigenvalue may be non-simple or its eigenvector non-positive, and criticality must be analyzed class by class.