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Three widely used species interaction models, the Nicholson-Bailey and Hassell-Varley host-parasite models and the Lotka-Volterra predator-prey model, are extended into space as well as time. One type of equilibrium having the same number of individuals at all locations is shown to exist analytically for the "symmetric dispersal" case. Another kind of equilibrium having different numbers of individuals at different locations in two-dimensional space is demonstrated by simulation. Stability analysis suggests that space-time models having symmetric dispersal are no more stable near equilibrium than the simple time models upon which they are based. In many situations the space-time models are less stable than the simple time models. This result can be reconciled with the experimental literature if one assumes a "stochastic persistence" effect which predicts that expected extinction time will increase with the number of independent subpopulations.

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN (·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

This paper deals with the following initial value problem: (1) ∂u/∂t−Δu+u p=0 on RN, N≥1, 0 0 and α≥0. For any x∈RN, the extinction time of x is defined as follows: TE(x)=sup{t>0,u(x,t)>0}, u(x,t) being the solution of (1). The following results are established. (i) If u0(x)≤A(|x|)+B|x−a|2/(1−p) for some a∈RN where A(|x|)=o(|x| 2/(1−p)) as |x|→+∞ and 0≤B TE(y). (ii) It is assumed that N=1. Let u(x,t) be the solution of (1). For a convenient behaviour of u0(x) as |x|→+∞, then u(x,t) tends to a limit as t→+∞, uniformly in compact sets on R. (iii) Let u0(r) be a nonnegative radial function satisfying (2). Then the solution of (1) with initial value u0(r) is radially symmetric and for a convenient behaviour of u0(r) as r→+∞, u(r,t) tends to a limit as r→+∞, uniformly on compact sets in R. For N≥2, the authors restrict themselves to the radial case.

The three phase induction motor, fed by inverse-parallel thyristor-diode pairs, shows self-excitation at 50/2Hz =25Hz subharmonic pulsating frequency in the region of plugging operation. The self-excitation can be verified also by exact calculation, however by this way it is hard to understand the physical phenomena, and long computation time is required. Present paper investigates the above phenomena by applying an equivalent circuit closely analogue to that of the induction machine, consisting of a three-phase resistance, inductance and voltage source. As it is proved in the paper, self-excitation is realized by the modification of extinction time. In the calculation the method of small deviations around an operatpoint is used. The boundary curve of the unstable region calculated by using the approximate equivalent circuit is in close agreement to these results obtained from exact calculations and test.

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure. The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.

Summary Several authors have studied and derived bounds for the mean extinction time of a discrete branching process. In this note, we obtain the mean and variance of the extinction time, given extinction of a continuous time Markov branching process. We also obtain bounds for the mean extinction time in terms of moments of the offspring distribution.

Properties of water spray injected into compressed atmospheres were studied in order to design an effective fixed spray system for use in a hyperbaric chamber. Fire extinction tests were conducted in such atmospheres with water spray from single nozzles. Extinction tests with nozzles fixed at various locations in the pressure vessel will be reported in Part II.

SUMMARY: Alterations in peptone concentration, pH and time of autoclaving of peptone water resulted in considerable variation in extinction time estimates. The longest extinction times were obtained with the higher peptone concentration, pH 7·0 and the shorter period of autoclaving.

Extinctions which occurred when liquids burning in open vessels were cooled by water sprays projected downwards on to the liquid surface have been correlated in one equation relating the extinction time with the spray and the fire properties. The extinction time was directly proportional to the drop size of the spray and the preburn time and inversely proportional to the rate of flow of water per unit of fire area, and also to the difference in temperature between the fire point and ambient temperature raised to the power 1·75. The equation does not cover conditions in which splash fires were developed. It is inferred that in the tests on which the equation is based cooling took place as a result of heat transfer between the oil and the water drops.

The extinction time of the radio signal, as the Mariner 9 spacecraft was occultated by Mars, together with an accurate ephemeris of the spacecraft were used to determine radii from the mass center to the occulting feature. Similarly estimations were made of the radius to a point where the pressure reached a certain fixed value. Several simple models were proposed to fit both sets of radii data.

Summary A new method for the determination of extinction time data is described. The method has been employed for the disinfection of Bact. coli by aqueous solutions of phenol over a range of concentrations and death times ranging from 2 to over 100 minutes. The use of extensive replication, short sampling intervals, and constant sample volumes withdrawn immediately after inoculation of the bactericide are among the chief features of the technique. Estimates of the mean extinction time obtained from the data are of reproducibility within limits sufficiently close to invalidate other extinction time methods, and are of comparable value with those obtained by the use of any other techniques. 5 The method has been satisfactorily applied to systems containing water-insoluble phenols solubilised in solutions of soaps (Berry and Bean9) where clumping of the test organisms might completely invalidate other methods of estimation of bactericidal activity.

The spatial arrangement of habitat patches in a metapopulation and the dispersal connections among them influence metapopulation persistence. Metapopulation persistence emerges from a dynamic process, namely the serial extinctions and recolonizations of local habitat patches, while measures of persistence are typically based solely on structural properties of the spatial network (e.g., spatial distance between sites). Persistence estimators based on static properties may be unable to capture the dynamic nature of persistence. Understanding the shape of the distribution of extinction times is a central goal in population ecology. Here, we examine the goodness of fit of the power law to patch persistence time distributions using data on a foundational metapopulation system-the Glanville fritillary butterfly in the angstrom land islands. Further, we address the relationship between structural measures of metapopulation persistence (i.e., metapopulation capacity) and our temporal distributional fits to patch persistence times based on a power law. Patch persistence time distributions were well fit by a power law for the majority of semi-independent networks. Power law fits to persistence time distributions were related to metapopulation capacity, linking structural and temporal measures of metapopulation persistence. Several environmental variables and measures of network topology were correlated with both measures of metapopulation persistence, though correlations tended to be stronger for the structural measure of metapopulation persistence (i.e., metapopulation capacity). Together, our findings suggest that persistence time distributions are useful dynamic properties of metapopulations, and provide evidence of a relationship between metapopulation structure and metapopulation dynamics.

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process. MARKOV BRANCHING PROCESS; EXTINCTION TIME; LOG-CONCAVITY; LOGCONVEXITY

The hypothesis that extinction events have recurred periodically over the last quarter billion years is greatly strengthened by new data on the stratigraphic ranges of marine animal genera. In the interval from the Permian to Recent, these data encompass some 13,000 generic extinctions, providing a more sensitive indicator of species-level extinctions than previously used familial data. Extinction time series computed from the generic data display nine strong peaks that are nearly uniformly spaced at 26 Ma intervals over the last 270 Ma. Most of these peaks correspond to extinction events recognized in more detailed, if limited, biostratigraphic studies. These new data weaken or negate most arguments against periodicity, which have involved criticisms of the taxonomic data base, sampling intervals, chronometric time scales, and statistical methods used in previous analyses. The criticisms are reviewed in some detail and various new calculations and simulations, including one assessing the effects of paraphyletic taxa, are presented. Although the new data strengthen the case for periodicity, they offer little new insight into the deriving mechanism behind the pattern. However, they do suggest that many of the periodic events may not have been catastrophic, occurring instead over several stratigraphic stages or substages.

Publisher Summary This chapter discusses the classes of compounds of current interest that have a reasonably sucrose-like sweet taste and are judged to have potential for sucrose replacement in food. The subject of design of nonnutritive sweeteners has been comprehensively discussed in the chapter. Sweetness potency (P), relative to a sucrose reference (R), is the most commonly reported descriptor for sweet compounds. It is strongly and inversely dependent on the concentration of R employed. In this chapter, P is indicated relative to use the levels (5-15%) of R whenever possible and reported as P(X)/R(Y) where X is the value of P and Y=%R. Sensory analytical procedures for obtaining such potency and sweet to non-sweet estimates, as well as their application have been reported. Sensory analytical procedures for the estimation of appearance time (AT) and extinction time (ET) in experimental sweeteners have been discussed in the chapter. In essence, it is believed that the sweet taste response is induced following the efficacious interaction of an active substance with a receptor protein located on the external periphery of a taste cell. The chapter discusses that the rationalization of the sweet taste of the many classes of sweet compounds by the existence of a common essential pharmacophore is not possible but more reasonable approach is the rationalization of common activity of the structurally diverse sweetener classes is the suggestion of multiple receptors; that the members of the same class of sweeteners may bind to the same receptor functionality. The topic on how to make the nonnutritive sweeteners safe—by rendering the sweeteners non-absorbable through the GI tract wall—has also been discussed in the chapter.

Neohesperidin dihydrochalcone (NHDHC), known since 1963 as an intensely sweet compound, is determined to be 340 +/- 60 (p less than 0.05) times more potent than sucrose. The unusual temporal properties of this material are hypothesized as being due to the effects of metabolism, conformation, chelation, or hydrophobicity. Forty-four analogues are synthesized to test the four hypotheses, none of which are strongly supported. A method of quantitation of temporal characteristics of tastant molecules is developed so as to allow comparison of taste appearance time (AT) and extinction time (ET) of experimental compounds. Four of the new compounds, 40 and 43-45, exhibit high sweetness potencies, ranging from 280 and 440 times sucrose, and may be useful in selected food systems. The temporal taste characteristics remain unimproved over NHDHC, however.

The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.

The histamine skin test, as modified by Giacovazzo, was carried out on 20 patients with cluster headache and on 20 control subjects, 10 of whom suffered from migraine and 10 who were healthy volunteers not presenting any form of headache. The test was carried out before and after 30 days treatment with timed-release dihydroergotamine methanesulphonate in a dose of 10 mg/day. Data recorded on a “headache time chart” before and after treatment were evaluated by statistical analysis. Block variance analysis was used to evaluate results of three parameters (latency time, hyperemia diameter, extinction time) emerging from the histamine skin test before and after treatment. A statistical comparison was also made of data obtained in basal conditions in the three groups of subjects, in order to confirm the usefulness of this simple test in the diagnosis of cluster headache. Les auteurs ont effectué le test cutané à l'histamine, d'après la méthode Giacovazzo, sur 20 personnes souffrant de céphalée de Horton, et sur 20 autres, parmi lesquelles 10 souffrant de migraines et les autres, volontaires, n'étant affectés d'aucun type de céphalée. Le test a été réalisé avant et après un traitement à base de 10 milligrammes par jour d'une préparation de dihydroergotamine métasulfonée se libérant d'une façon programmée, et ce durant 30 jours consécutifs. Les résultats enregistrés sur la “fiche des relevés horaires de la céphalée” ont été analysés statistiquement avant et après le traitement thérapeutique. C'est ensuite qu'ont été analysés, grâce à la méthode de variance à blocs (phases pré et post thérapie), les trois paramètres déduits du test cutané à l'histamine (temps de latence, diamètre d'hyperhémie, temps d'extinction). Et, il a finalement été possible, à l'appui de la validité diagnostique de ce simple test, de confronter statistiquement les résultats des trois paramètres de base relatifs aux différents groupes de patients. E’ stato eseguito il test cutaneo all'istamina, secondo Giacovazzo, in 20 pazienti di cefalea a grappolo e in 20 soggetti di controllo, dei quali 10 affetti da emicrania e 10 volontari sani. Il test è stato eseguito prima e dopo trattamento con 10 mg al dì, per trenta giorni, di un preparato di diidroergotamina metansulfonato a liberazione programmata. Sono stati analizzati statisticamente i dati dedotti dalla “carta oraria della cefalea”, prima e dopo trattamento terapeutico. Il metodo della varianza per blocchi è stato usato per confrontare (fasi pre e post terapia) i tre parametri dedotti dal test cutaneo all'istamina (tempo di latenza, diametro di iperemia, tempo di estinzione). Un ulteriore confronto statistico è stato infine ese-guito tra i dati desunti dai tre parametri in condizioni di base, nei vari gruppi di pazienti, a conforto della validità diagnostica di questo semplice test.

In recent years a great deal of attention has been given to the Cluster Headache (CH) by various groups of researches, due to the peculiarity of this form of primary headache from a clinical as well as from a hormonal point of view [4, 8]. The ethio-pathogenesis of the disease is not known as yet.

A simple Poisson process, [N(t), t ≥ 0], with parameter λ, is a random process with stationary independent increments such that if 0 ≤ t1 < t2, then $$ {\rm{Pr[N(}}{{\rm{t}}_{\rm{2}}})\; - \;{\rm{N(}}{{\rm{t}}_{\rm{1}}}{\rm{)}}\;{\rm{ = k] = [\lambda (}}{{\rm{t}}_{\rm{2}}} - {{\rm{t}}_1}){]^{\rm{k}}}\;\exp [ - {\rm{\lambda (}}{{\rm{t}}_{\rm{2}}} - {{\rm{t}}_{\rm{1}}})]/{\rm{k! k = 0,1,2}}.. $$ (1)

Neohesperidin dihydrochalcone (NHDHC), known since 1963 as an intensely sweet compound, is determined to be 340 +/- 60 (p less than 0.05) times more potent than sucrose. The unusual temporal properties of this material are hypothesized as being due to the effects of metabolism, conformation, chelation, or hydrophobicity. Forty-four analogues are synthesized to test the four hypotheses, none of which are strongly supported. A method of quantitation of temporal characteristics of tastant molecules is developed so as to allow comparison of taste appearance time (AT) and extinction time (ET) of experimental compounds. Four of the new compounds, 40 and 43-45, exhibit high sweetness potencies, ranging from 280 and 440 times sucrose, and may be useful in selected food systems. The temporal taste characteristics remain unimproved over NHDHC, however.

There are problems in population biology and in neurobiology that require the evaluation of first-passage-time probability density functions (p.d.f.) for diffusion or for Gaussian processes. For instance, the fixation of a gene may be viewed as the process leading the frequency of that gene to attain the value one for the first time; the extinction time of a population can be interpreted as the time when first the population size shrinks to some preassigned small level; the firing of a neuron can at times be modeled as the first crossing through a critical threshold value by the random process depicting the time course of the membrane potential. The literature on this subject is so vast that it cannot and need not be referenced here. It may suffice to mention that an explicit reference to these problems can be found in Ricciardi (1977) and in the literature quoted therein. Instead, we would like to stress that the determination of first-passage-time densities and of its features is in general a formidable task. It is fair to claim that so far only computational methods have been employed, which require the use of large scale computers, are extremely time consuming and hence not quite suitable to carry out the systematic analyses imposed by the nature of the biologically interesting problems that usually involve numerous parameters to be specified by best fitting procedures (Favella et al. 1982) Ricciardi and Sato, 1983a; Ricciardi et al., 1983) .

The inactivation of coliphage T6r by phenol has been examined using an extinction method and the reproducibility of the results shown to be acceptable. Two methods of interpreting the results are described and their relationship discussed. A linear relationship has been found between log extinction time and log phenol concentration for extinction times between 2 and 200 min.

A multitype branching process with immigration is considered. A necessary and sufficient condition is found for when the process has a limiting stationary distribution. As a corollary to this result we are able to give necessary and sufficient conditions for when the extinction time for an ordinary multitype process has finite expectation.

If $\{Z_n\}$ is a Galton-Watson process with mean one, and $\tau$ is the extinction time, it is shown that $EZ^{1+\alpha} < \infty$ implies $E\tau^\beta = \infty$ for $\beta > 1/\alpha, 0 < \alpha < 1$. Conditions which imply $EZ^{1+\alpha} = \infty$ and $E\tau^\beta < \infty$ for $\beta < 1/\alpha, 0 < \alpha < 1$ are given. Necessary and sufficient conditions for $EX^{m+\alpha} < \infty$ or $EX^m \log X < \infty$ are given in terms of the Laplace transform of a general nonnegative random variable $X, 0 < \alpha < 1, m = 0, 1,\cdots$.

This study aimed to investigate the clinical significance of contrast enhanced ultrasound (CEUS) in diagnosis of benign prostatic hyperplasia (BPH) through comparing CEUS parameters between BPH and normal person. A retrospective study of sixty BPH patients (aged 73.5 ± 20.5 years old) and thirty normal controls without prostate diseases (aged 75.3 ± 19.7 years old) who had accepted CEUS detection were performed. Time-intensity curves were obtained for all tests in regions of interest. Images were processed using ACQ software and the following parameters were obtained: arrival time (AT), peak intensity (P), time to peak (TP), area under the curve (AUC), mean transit time (MTT) and extinction time (ET). Differences in inner and outer gland of prostate between BPH and the normal tissue were evaluated. There was a clear boundary between the inner and outer gland of BPH prostate. AT, TP, MTT, ET and P in BPH outer gland were significantly higher than the control group. In inner gland, MTT, ET, AUC and P were also significantly higher than the controls. The accurate rate to diagnose BPH using CEUS was 95.6%, and the sensitivity and specificity were 95.0% and 96.7%, respectively. Among these significantly changed parameters, the increases of MTT, ET and AUC in inner gland and AT, TP in outer gland were most likely related to BPH. These parameters provide an objective visual assessment to diagnosis of BPH. The virtual slide(s) for this article can be found here: http://www.diagnosticpathology.diagnomx.eu/vs/4852383312229155

RAPPORT TECHNIQUE

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population., We review the use and the derivation of continuum limits in the context of ecology and evolution and provide key analytic expressions of the fixation probability, the mean extinction time and the stationary distribution. By example, we compute these quantities for the Wright–Fisher process and the stationary distribution of the logistic growth equation.

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

An attempt was made to replicate Sheffield & Roby’s (1950) findings that saccharin consumption did not extinguish when the saccharin was presented in the absence of a primary reinforcer. The present study reduced the number of prior eating experiences by the Ss to 1/8 and increased the amount of extinction time to 17 times that of Sheffield and Roby’s Ss. Safeguards were also taken to insure that there was no food present in the S’s stomach in order to meet a criticism leveled at Sheffield and Roby’s study by Smith & Capretta (1956). Despite the two modifications to Sheffield and Roby’s design, the present study replicated Sheffield and Roby’s findings. Saccharin consumption did not extinguish during 4,380 h of extinction time. This suggests that Sheffield and Roby’s findings are still perhaps the best single argument against the “secondary reinforcement” explanation of saccharin consumption.

The influence on response rate during extinction of the reinforcement schedule during acquisition and the time interval between acquisition and extinction was studied in a 2 by 2 factorial design with each condition represented by 10 female albino rats. Two groups were continuously reinforced and 2 groups were intermittently reinforced on a graduated schedule terminating in a fixed ratio-5 schedule at the end of acquisition. Under each reinforcement contingency, 1 subgroup was extinguished 24 hr. after acquisition and another 3 weeks after acquisition. An analysis of variance on the rate of responding in the first 35 min. of extinction revealed significant influences attributable to reinforcement schedule and to the interaction of schedule with time interval; the latter arising from a greater response rate with increased acquisition to extinction time interval following the continuous reinforcement contingency and a lesser rate following the intermittent contingency. The results are interpreted in terms of changes over time in the stimulus generalization gradients of the stimuli controlling the behavior chains associated with the 2 reinforcement schedules.

The theory of extreme values is shown to be relevant in the analysis of two stochastic biological phenomena.