Extinction times for a general birth, death and catastrophe process

The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Extinction times for a birth-death process with two phases

Many populations have a negative impact on their habitat or upon other species in the environment if their numbers become too large. For this reason they are often subjected to some form of control. One common control regime is the reduction regime: when the population reaches a certain threshold it is controlled (for example culled) until it falls below a lower predefined level. The natural model for such a controlled population is a birth-death process with two phases, the phase determining which of two distinct sets of birth and death rates governs the process. We present formulae for the probability of extinction and the expected time to extinction, and discuss several applications. (c) 2006 Elsevier Inc. All rights reserved.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Gauge Poisson representations for birth/death master equations

Poisson representation techniques provide a powerful method for mapping master equations for birth/death processes -- found in many fields of physics, chemistry and biology -- into more tractable stochastic differential equations. However, the usual expansion is not exact in the presence of boundary terms, which commonly occur when the differential equations are nonlinear. In this paper, a gauge Poisson technique is introduced that eliminates boundary terms, to give an exact representation as a weighted rate equation with stochastic terms. These methods provide novel techniques for calculating and understanding the effects of number correlations in systems that have a master equation description. As examples, correlations induced by strong mutations in genetics, and the astrophysical problem of molecule formation on microscopic grain surfaces are analyzed. Exact analytic results are obtained that can be compared with numerical simulations, demonstrating that stochastic gauge techniques can give exact results where standard Poisson expansions are not able to.

Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

Stochastic models for mainland-island metapopulations in static and dynamic landscapes

This paper has three primary aims: to establish an effective means for modelling mainland-island metapopulations inhabiting a dynamic landscape: to investigate the effect of immigration and dynamic changes in habitat on metapopulation patch occupancy dynamics; and to illustrate the implications of our results for decision-making and population management. We first extend the mainland-island metapopulation model of Alonso and McKane [Bull. Math. Biol. 64:913-958,2002] to incorporate a dynamic landscape. It is shown, for both the static and the dynamic landscape models, that a suitably scaled version of the process converges to a unique deterministic model as the size of the system becomes large. We also establish that. under quite general conditions, the density of occupied patches, and the densities of suitable and occupied patches, for the respective models, have approximate normal distributions. Our results not only provide us with estimates for the means and variances that are valid at all stages in the evolution of the population, but also provide a tool for fitting the models to real metapopulations. We discuss the effect of immigration and habitat dynamics on metapopulations, showing that mainland-like patches heavily influence metapopulation persistence, and we argue for adopting measures to increase connectivity between this large patch and the other island-like patches. We illustrate our results with specific reference to examples of populations of butterfly and the grasshopper Bryodema tuberculata.

Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

Anatomy of the pharyngeal jaw apparatus of Zenarchopterus (Gill) (Teleostei : Beloniformes)

The structure of the pharyngeal jaw apparatus (PJA) of Zenarchopterus dispar and Z. buffonis, carnivorous estuarine and freshwater West-Pacific halfbeaks, was investigated using dissection, light, and scanning electron microscopy as part of a comparison with estuarine and marine herbivorous confamilials. The Zenarchopterus PJA differs from published descriptions of hemiramphid PJAs in that the otic capsules are less pronounced; the pharyngocranial articulation facet is trough-like; the third pharyngobranchials are ankylosed; the second pharyngobranchial anterior processes are relatively hypotrophied; all pharyngeal teeth except the posterior teeth in the fifth ceratobranchial face posteriorly; the muscularis craniopharyngobranchialis 2 posterior is short; the muscularis craniopharyngobranchialis 2 anterior is lacking, as is its insertion site, the inferior parasphenoid apophysis; the protractor pectoralis is well developed; the pharyngocleithralis internus originates dorsal to the level of the fifth ceratobranchial bony process; the fifth ceratobranchial bony processes are directed ventrolaterally; the opposing upper and lower tooth fields appear not to occlude erosively; and the muscular portion of the pharyngohyoideus is well developed anteriorly. The extent of these differences and their implications for the function of the PJA support recent molecular studies that suggest that the Hemiramphidae is polyphyletic. (C) 2004 Wiley-Liss, Inc.

Subcellular localization determines MAP kinase signal output

The Raf-MEK-ERK MAP kinase cascade transmits signals from activated receptors into the cell to regulate proliferation and differentiation. The cascade is controlled by the Ras GTPase, which recruits Raf from the cytosol to the plasma membrane for activation. In turn, MEK, ERK, and scaffold proteins translocate to the plasma membrane for activation. Here, we examine the input-output properties of the Raf-MEK-ERK MAP kinase module in mammalian cells activated in different cellular contexts. We show that the MAP kinase module operates as a molecular switch in vivo but that the input sensitivity of the module is determined by subcellular location. Signal output from the module is sensitive to low-level input only when it is activated at the plasma membrane. This is because the threshold for activation is low at the plasma membrane, whereas the threshold for activation is high in the cytosol. Thus, the circuit configuration of the module at the plasma membrane generates maximal outputs from low-level analog inputs, allowing cells to process and respond appropriately to physiological stimuli. These results reveal the engineering logic behind the recruitment of elements of the module from the cytosol to the membrane for activation.

Procedures and parameters in the real-time program refinement calculus

The real-time refinement calculus is a formal method for the systematic derivation of real-time programs from real-time specifications in a style similar to the non-real-time refinement calculi of Back and Morgan. In this paper we extend the real-time refinement calculus with procedures and provide refinement rules for refining real-time specifications to procedure calls. A real-time specification can include constraints on, not only what outputs are produced, but also when they are produced. The derived programs can also include time constraints oil when certain points in the program must be reached; these are expressed in the form of deadline commands. Such programs are machine independent. An important consequence of the approach taken is that, not only are the specifications machine independent, but the whole refinement process is machine independent. To implement the machine independent code on a target machine one has a separate task of showing that the compiled machine code will reach all its deadlines before they expire. For real-time programs, externally observable input and output variables are essential. These differ from local variables in that their values are observable over the duration of the execution of the program. Hence procedures require input and output parameter mechanisms that are references to the actual parameters so that changes to external inputs are observable within the procedure and changes to output parameters are externally observable. In addition, we allow value and result parameters. These may be auxiliary parameters, which are used for reasoning about the correctness of real-time programs as well as in the expression of timing deadlines, but do not lead to any code being generated for them by a compiler. (c) 2006 Elsevier B.V. All rights reserved.

Relative predictive value of intraventricular dysynchrony in determining long-term outcome in patients with known or suspected coronary disease

Improvement of intra-ventricular dysynchrony (IVD) in pts undergoing bi-ventricular pacing is associated with clinical improvementbut little isknownabout the relationship between IVD and prognosis.We sought whether IVD influences long-term outcome in pts with known or suspected coronary disease (CAD). Tissue Doppler imaging was performed in 184 pts (aged 61±10 years, 67% male) prior to dobutamine echo. From velocity curves the interval between QRS onset and max systolic velocity (Ts) was measured in basal septal, lateral, inferior and anterior segments. The maximal difference in Ts between segments (TsMax) was used as a measure of IVD. The standard deviation (TsSD) between all segments and the septal-lateral difference (TsSL) were also calculated. Pts were followed up for a median interval of 5 years and a Cox model used for survival analysis. The medianwall motion index (WMI) was 1.3 (IQR 1.0–1.8) at rest and 1.4 (IQR 1.3–1.9) at stress. The table shows IVD parameters. Forty-one deaths occurred during follow-up. Pts who died during follow-up, compared to survivors, showed greater IVD. WMI at rest (p = 0.03) and peak stress (p = 0.02), TsSD (p = 0.06), TsSL (p = 0.02) and TsMax (p = 0.05) but not QRS width were univariate predictors of mortality. TsSL was the only independent predictor of death (p = 0.01). Therefore, IVD is common in pts with known or suspected CAD. Pts with more IVD have reduced long-term survival, independent of WMI.

How many variables can humans process?

The conceptual complexity of problems was manipulated to probe the limits of human information processing capacity. Participants were asked to interpret graphically displayed statistical interactions. In such problems, all independent variables need to be considered together, so that decomposition into smaller subtasks is constrained, and thus the order of the interaction. directly determines conceptual complexity. As the order of the interaction increases, the number of variables increases. Results showed a significant decline in accuracy and speed of solution from three-way to four-way interactions. Furthermore, performance on a five-way interaction was at chance level. These findings suggest that a structure defined on four variables is at the limit of human processing capacity.

Towards platform-independent real-time systems

Real-time software systems are rarely developed once and left to run. They are subject to changes of requirements as the applications they support expand, and they commonly outlive the platforms they were designed to run on. A successful real-time system is duplicated and adapted to a variety of applications - it becomes a product line. Current methods for real-time software development are commonly based on low-level programming languages and involve considerable duplication of effort when a similar system is to be developed or the hardware platform changes. To provide more dependable, flexible and maintainable real-time systems at a lower cost what is needed is a platform-independent approach to real-time systems development. The development process is composed of two phases: a platform-independent phase, that defines the desired system behaviour and develops a platform-independent design and implementation, and a platform-dependent phase that maps the implementation onto the target platform. The last phase should be highly automated. For critical systems, assessing dependability is crucial. The partitioning into platform dependent and independent phases has to support verification of system properties through both phases.