Göran Högnäs

Approximation of delay systems—a case study

Abstract The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...

Population extinction in discrete-time stochastic population models with an Allee effect

The Allee effect is often modeled as a threshold level below which there is population extinction. A minimum population size is required for population persistence. The Allee effect has been primarily studied in deterministic models. In this investigation, we develop two discrete-time stochastic population models with an Allee effect. The stochastic models are discrete-time Markov chains with demographic stochasticity. They are based on two well-known deterministic discrete-time population models. It is shown that the conditional means of the stochastic models follow the solution dynamics of the underlying deterministic models. However, near the Allee threshold value, the dynamics of the deterministic and stochastic models differ significantly. The first model is based on the well-known Ricker population model with an Allee effect. The second model is more complex. The population is subdivided according to mating status of adults and is based on a model developed for the northern spotted owl. The dynamics of the underlying deterministic models are reviewed first, then the stochastic models are formulated. The stochastic models are new formulations. Our goal is to study the probability of population extinction in the stochastic models.

Population models with environmental stochasticity

Two discrete population models, one with stochasticity in the carrying capacity and one with stochasticity in the per capita growth rate, are investigated. Conditions under which the corresponding Markov processes are null recurrent and positively recurrent are derived.

On the quasi-stationary distribution of a stochastic Ricker model

We model the evolution of a single-species population by a size-dependent branching process Zt in discrete time. Given that Zt = n the expected value of Zt+1 may be written nexp(r - [gamma]n) where r > 0 is a growth parameter and [gamma] > 0 is an (inhibitive) environmental parameter. For small values of [gamma] the short-term evolution of the normed process [gamma]Zt follows the deterministic Ricker model closely. As long as the parameter r remains in a region where the number of periodic points is finite and the only bifurcations are the period-doubling ones (r in the beginning of the bifurcation sequence), the quasi-stationary distribution of [gamma]Zt is shown to converge weakly to the uniform distribution on the unique attracting or weakly attracting periodic orbit. The long-term behavior of [gamma]Zt differs from that of the Ricker model, however: [gamma]Zt has a finite lifetime a.s. The methods used rely on the central limit theorem and Markovs inequality as well as dynamical systems theory.

Random semigroup acts on a finite set

Let X be a finite set and S a semigroup of transformations of X. We investigate the trace on X of a random walk on S. We relate the structure of the trace process, which turns out to be a Markov chain, to that of the random walk. We show, for example, that all periods of the trace process divide the period of the random walk.

Stability classification of a Ricker model with two random parameters

We consider a stochastic version of the Ricker model describing the density of an unstructured isolated population. In particular, we investigate the effects of independently varying the per capita growth rate and the parameter governing density dependent feedback. We derive conditions on the distributions sufficient to guarantee different forms of stochastic stability such as null recurrence or positive recurrence. We find, for example, that null recurrence appears in two widely different scenarios: when there is a mean-zero growth rate or via a growth-catastrophe behaviour.

Using WinBUGS to study family frailty in child mortality, with an application to child survival in Ivory Coast

This article analyzes the effects of unobserved family heterogeneity in children survival times through a Bayesian approach. We rely on survey data from Ivory Coast and use a proportional hazard model with multiplicative random effect. With such a model, the usual assumption of independence of observations is avoided. The posterior distributions of the parameters are estimated through a Gibbs sampler algorithm using the WinBUGS software. This technique overcomes the possible local convergence problem observed with the commonly used Expectation-Maximization method.

A note on products of random matrices

Let P be a probability distribution on a set of d x d matrices. Let Pn denote the n-fold convolution of P with itself. Then tightness of the sequence Pn implies that the Cesaro sequence (1/n[sigma] Pn converges to an idempotent probability measure Q and the support of Q is exactly the set m(S) of matrices of minimal rank in the semigroup S generated by the support of P. Furthermore the set m(S) is a completely simple semigroup with compact group factor. The convergence of Pn can be characterized in terms of the Rees-Suschkewitsch decomposition of m(S).

Special issue on stochastic difference equations

As we all know, there has been a virtual explosion of interest in nonlinear dynamical systems during the last few decades. Studies and experiments with difference equations have inspired the creation of a whole new branch of mathematics, chaos theory, where paradigmatic examples are given by systems generated by very simple low-dimensional maps. Apart from being a beautiful mathematical theory, chaos theory has made a lasting impression in applied fields by introducing a host of new and important, often unifying concepts. Our point of departure is a discrete-time dynamical system expressible as a difference equation

A mixed random walk on nonnegative matrices: A law of large numbers

In this paper a mixed random walk on nonnegative matrices has been studied. Under reasonable conditions, existence of a unique invariant probability measure and a law of large numbers have been established for such walks.