Averaging in random systems of nonnegative matrices

It is proved that for the top Lyapunov exponent of a random matrix system of the form $\{A D(\omega)\}$, where $A$ is a nonnegative matrix and $D(\omega)$ is a diagonal matrix with positive diagonal entries, is bounded from below by the top Lyapunov exponent of the averaged system. This is in contrast to what one should expect of systems describing biological metapopulations.


Introduction
We assume throughout that (Ω, F , P) is a probability space: F is a σ-algebra of subsets of Ω, and P is a probability measure defined on F .
By a random matrix system we understand a measurable family S : Ω → R N ×N , A random matrix system gives rise to a discrete-time linear (skew-product) random (semi)dynamical system on Ω × R N consisting of iterates of the vector bundle morphism (ω, u) → (θω, S(ω)u), ω ∈ Ω, u ∈ R N .
The following result is a part of the Furstenberg-Kesten theorem (see, e.g., [1,Thm. 3.3.3]). Proposition 1. For a random matrix system (1) assume that the mapping ln + S(·) belongs to L 1 (Ω, F , P). Then there exists λ ∈ [−∞, ∞) such that for a.e. ω ∈ Ω the equality Moreover, (Here and in the sequel · denotes the Euclidean matrix or vector norm, depending on the context.) We will call λ as above the top Lyapunov exponent of the random matrix system (1).
In the present paper we consider random matrix systems of a special form, namely such that S(ω) = AD(ω), where A is a constant (that is, independent of ω) matrix with nonnegative entries and D(ω) is a diagonal matrix with positive diagonal entries.
Such random matrix systems occur in modeling so-called metapopulations, that is, populations in which individuals live in N spatially separated patches (see, e.g., [9]). Here u i , 1 ≤ i ≤ N , is the number of individuals in patch i, d i is the fitness of an individual in patch i, and a ij is, for i = j, the fraction of the population from patch j that disperse to patch i.
The top Lyapunov exponent measures the overall fitness of the metapopulation: the larger it is the more viable the (meta)population should be. Indeed, if A is a primitive matrix (meaning that some of its powers has all entries positive) then the logarithmic growth rate of iterates of any positive vector equals the top Lyapunov exponent.
It is an important subject in population dynamics to analyze the influence of seasonal variations on the fitness. To quote Sebastian J. Schreiber [9]: Temporal fluctuations in environmental conditions can lead to fluctuations in population growth rates. For a given mean population growth rate, one expects that extinction risk increases with temporal variation in the growth rates.
Let us look at the mathematical interpretation of the above statement in the language of random matrix systems of the form S(ω) = AD(ω). As the dynamical system generated by θ on the base space is ergodic, for each patch i, 1 ≤ i ≤ N , Birkhoff's ergodic theorem states that for a.e. ω ∈ Ω the limit exists and equals the expected value which can be interpreted as the mean population growth rate in isolated patch i. As dispersal rates are independent of time, one compares the top Lyapunov exponent of the original system with the top Lyapunov exponent of the system with the population growth rate in each patch i replaced by its geometric mean. The latter Lyapunov exponent equals just the logarithm of the spectral radius of AD, whereD is the diagonal matrix obtained by taking the geometric means of the entries of D. Therefore, our expectations should be that the top Lyapunov exponent of the system with temporal variation is not larger than the metapopulation growth rate for the averaged growth rates in all patches.
However, our Theorem 2.1 shows that the reverse is true.
The paper is organized as follows. In Section 2 the main concepts are introduced and Theorem 2.1 is formulated. In Section 3 we give a proof of Theorem 2.1 under the assumption that A has all entries positive. Section 4 deals with a general case.

Main concepts
Assume that A is an N × N nonnegative matrix.
We consider random matrix systems of the form We thus have s ij (ω) = a ij d j (ω) for all 1 ≤ i, j ≤ N , ω ∈ Ω. As we will be using some results from [8], we introduce here some auxiliary functions from that paper, as well as their properties: where ln 0 = −∞, e −∞ = 0. As a consequence of (A2),d i ∈ [0, ∞). LetD stand for the diagonal matrix diag (d 1 , . . . ,d N ).
The matrix AD is nonnegative, so, by the Frobenius-Perron theorem, see [3, Thm. 1.3.2], its spectral radius is an eigenvalue such that an eigenvector corresponding to it can be chosen nonnegative.
The following is the main result of the paper. The case of the zero spectral radius of AD is obvious, so from now on we assume that the spectral radius of AD is positive.

A is a positive matrix
In the present section we give a proof of Theorem 2.1 under the additional assumption that A is a positive matrix. This allows us to apply the theory of random systems of positive matrices as presented in [8].
The top Lyapunov exponent λ can now be expressed as the logarithmic growth rate of some distinguished positive vector. Indeed, the following result holds.
For each i, by multiplying the above inequality by a ij , 1 ≤ j ≤ N , and adding the resulting inequalities one obtains, after some calculation, that it follows that lim n→∞ŵ i (n) =w i for all 1 ≤ i ≤ N . We have therefore found a positive vectorw such that where the inequality is meant to hold coordinatewise. By [3, Thm. 2.1.11], the spectral radius of AD does not exceed e λ , which concludes the proof.  For any ǫ > 0 denote by λ ǫ the top Lyapunov exponent of system (4) with A replaced by A + ǫB, that is, of the system The fact that λ equals the limit, as ǫ → 0 + , is a consequence, for instance, of [4, Thm. 1]. We will give here, however, a much more direct proof here. It is a standard exercise that for a nonnegative N by N matrix C there holds C = sup{ Cu : u = (u 1 , . . . , u N ), u i ≥ 0, u = 1 } (cf., e.g., [7, Lemma 3.1.1]). Consequently, for any 0 < ǫ 1 ≤ ǫ 2 we have ǫ2 (ω) , and, as a result, On the other hand, it follows from (3) that the top Lyapunov exponent is upper semicontinuous, in particular, λ ≥ lim sup ǫ→0 + λ ǫ . Therefore An analogous reasoning can be repeated for averaged matrices. Thus we obtain the desired result.