Perron-Frobenius Growth in Multitype Branching

From Mean Progeny to Long-Run Growth

In a multitype branching process, individuals of different types may have different expected numbers of children of each type. The mean progeny matrix collects these expectations into a single linear map. Its powers describe expected population counts across generations, and Perron–Frobenius theory identifies the dominant growth rate and type profile. One qualification matters: irreducibility gives a positive dominant eigenvector, but convergence of the generation-by-generation type proportions also requires ruling out periodic oscillation.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and consider a discrete-generation branching process \((Z_n)_{n\ge 0}\) with finite type set \(S=\{1,\ldots,d\}\). The population vector \(Z_n=(Z_n(1),\ldots,Z_n(d))\) is a row vector; \(Z_n(j)\) is the number of type-\(j\) individuals in generation \(n\). Each type-\(i\) individual produces a random vector of type counts, independently of other individuals and generations, with finite expected counts. Define \(M_{ij}\) to be the expected number of type-\(j\) children of one type-\(i\) parent. Thus \(M\) is a nonnegative \(d\)-by-\(d\) matrix, and the row-vector convention gives \(\mathbb{E}[Z_{n+1}\mid Z_n]=Z_nM\).

Irreducibility means every type can eventually lead to every other type through a sequence of positive-mean reproduction links. Primitivity is stronger: for some common generation count, every type can lead to every other type in exactly that many steps. In particular, a primitive matrix is irreducible, but an irreducible matrix can have a cyclic structure that prevents proportions from settling to a single limit.

We use the following standard form of the Perron–Frobenius theorem. Its proof is a substantial result in matrix theory and is not developed here. For an irreducible nonnegative matrix \(M\) with positive spectral radius \(r\), \(r\) is an eigenvalue with strictly positive right and left eigenvectors \(v\) and \(u\), so \(Mv=rv\) and \(uM=ru\). The eigenvalue \(r\) is simple, and every eigenvalue \(\lambda\) satisfies \(|\lambda|\le r\). If \(M\) is primitive, every other eigenvalue has modulus strictly less than \(r\). The positive eigenvalue \(r\) is called the Perron root. We use \(u\) as a row vector and \(v\) as a column vector.

Expected Counts and the Perron Root

The mean matrix acts one generation at a time. The next result makes precise why its powers are the right object, and requires only finite offspring means and the branching independence encoded in the model.

Proof. For a type-\(i\) parent, the expected vector of children is row \(i\) of \(M\). Conditional on the generation-\(n\) population, add the expected offspring vectors of all individuals. Linearity of conditional expectation gives \(\mathbb{E}[Z_{n+1}\mid Z_n]=Z_nM\). Taking expectations and using that \(M\) is a fixed finite matrix yields \(\mathbb{E}[Z_{n+1}]=\mathbb{E}[Z_n]M\). At \(n=0\), \(\mathbb{E}[Z_0]=x=xM^0\). If \(\mathbb{E}[Z_n]=xM^n\), the recurrence gives \(\mathbb{E}[Z_{n+1}]=xM^nM=xM^{n+1}\). Induction proves the statement for every \(n\). \(\square\)

The next result derives the asymptotic mean profile for the primitive case. The positive left eigenvector, not the right eigenvector, describes proportions when populations are written as row vectors.

Proof. The denominator \(uv\) is strictly positive because both eigenvectors have strictly positive entries. Normalize the eigenvectors so that \(uv=1\); this changes neither claimed limit. The matrix \(A=vu\) then satisfies \(A^2=v(uv)u=A\), \(MA=rA\), and \(AM=rA\). Define \(C=M-rA\). These identities give \(AC=CA=0\), and, for every \(n\ge1\), \(M^n=r^nA+C^n\).

Every eigenvalue of \(C\) other than the removed Perron eigenvalue has modulus strictly less than \(r\), by the primitive case of Perron–Frobenius. In finite dimensions, the Jordan form expresses each entry of \(C^n\) as a finite sum of terms bounded by constants times \(n^k|\lambda|^{n-k}\), for eigenvalues \(\lambda\) of \(C\) and finite integers \(k\). Dividing such terms by \(r^n\) makes them tend to zero: for \(|\lambda|<r\), a fixed polynomial factor in \(n\) is dominated by the geometric decay \((|\lambda|/r)^n\). Hence \(M^n/r^n\to A=vu\). Multiplying on the left by \(x\) gives \(xM^n/r^n\to (xv)u\) under the chosen normalization. For general normalization this is \((xv)/(uv)\,u\).

Since \(x\) is nonzero and nonnegative and \(v\) is strictly positive, \(xv>0\). Summing the coordinates of the first limit therefore gives a positive limiting constant times \(\sum_j u_j\). Dividing the vector limit by its coordinate sum proves the second limit. Finally, the total expected population is asymptotic to a positive constant times \(r^n\); taking \(n\)-th roots shows that its exponential growth rate is \(r\). \(\square\)

This conclusion concerns expected counts, not automatically the realized random population \(Z_n\). A claim about almost-sure growth or random limiting type proportions requires additional hypotheses and a branching-process limit theorem. The matrix calculation alone establishes the mean behavior.

Why Irreducibility Alone Is Not Enough

For a primitive matrix the spectral gap makes all non-Perron contributions negligible relative to \(r^n\). An irreducible matrix can instead have other eigenvalues of modulus \(r\). Their contributions need not disappear, so the expected type proportions can oscillate even though the overall population has a well-defined exponential growth rate.

Common mistake. Using the right Perron eigenvector as the type proportions for a row population vector can give the wrong answer. In the second example, the right eigenvector is \((1,1)^{\mathsf T}\), which might suggest equal proportions. But the row-vector recurrence is \(xM^n\), and the limiting proportions are given by the left eigenvector: \((1,2)/3=(\tfrac13,\tfrac23)\). Substitution verifies the distinction: \((1,2)M=(4,8)=4(1,2)\), whereas the right eigenvector describes how \(M\) acts on column vectors.

The practical workflow is therefore: form the mean progeny matrix using the stated row convention; check irreducibility; find the Perron root and positive left and right eigenvectors; and check primitivity before claiming convergence of generation-by-generation expected type proportions. The Perron root governs exponential growth of the mean population, while primitivity guarantees a single stable expected type profile. If periodicity remains, growth can still be identified while proportions cycle across generations.