A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.

This paper presents a simple model of the decision problems which face many animals in the wild. Animals have an upper limit to the amount of energy reserves which can be stored as body fat. They constantly use energy and find food as a stochastic process. Death occurs if energy reserves fall to zero or if the animal is taken by a predator. Typically animals have a range of behavioural options which differ in the distribution of food gained and in the associated predation risk. There is often a trade-off between starvation and predation in that animals have to expose themselves to higher predation risks in order to gain more food.

The above situation is modelled as a finite-state, finite-time-horizon Markov decision problem. The policy which maximises long-term survival probability is characterised. As special cases two trade-offs are analysed. It is shown that an animal should take fewer risks in terms of predation as its reserves increase, and that an animal should reduce the variability of its food supply at the expense of its mean gain as its reserves increase.

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

Consider a set S of countable non-negative matrices satisfying the property that for any two indices i, j, for some n ≧ 1 there are matrices M1, M2, · · ·, Mn in S with (M1M2 · · · Mn)ij >0. For non-negative vectors x set Tx = supM∈SMx, where the supremum is taken separately in each coordinate. Assume that for each x with Tx finite in each coordinate there is a matrix in S which achieves the supremum simultaneously for all coordinates. With these two assumptions on S, the R-theory for a countable irreducible matrix is extended to the operator T. The results are used to consider the existence of stationary optimal policies for Markov decision processes with multiplicative rewards.

A max min formula for the spectral radius of a non-negative matrix is derived from a characterization of non-singular M-matrices in terms of diagonal stability.

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.

The concept of a stable population, which is of basic importance in demography, is generalized to person-flow models as defined by Gani. Stability and asymptotic stability of manpower systems are investigated in terms of the input process and the transition structure. Extending Bartholomew ((1973), Chapter 3) we obtain a result on the limiting behaviour of Gani-type models with asymptotic exponential input processes. The linear population-dynamic model of Leslie (1945) may be considered as a further case.

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij of the total amount of flow which arrived at node i during the previous time period. The fraction pij whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij are discussed.