Asymptotic Extinction Research Articles

Population of entities with three individual states and asymmetric interactions

In various applications in life and social sciences the agents of a complex system, which can be individuals of a sub–populations, can have one of the few inner states or can choose one of few strategies. Their states are effects of interactions with other agents. Behavior is described by a kinetic–like a nonlinear equation. In the present paper, we study the behavior of solutions in the case of three possible states and show that in some cases a kind of self–organization occurs but in others, a periodic behavior characterizes the system. We can observe a wide variety of dynamics that can relate to the behavior of real systems. The model contains a natural interaction intensity parameter γ≥1. Case γ>1 leads to “very” nonlinear structures. We prove that the system has no non–constant periodic solutions. We propose conditions guaranteeing the asymptotic stability of equilibrium points on the boundary that corresponds to the asymptotic extinction of two states. Moreover, conditions for the uniqueness and instability of an inner equilibrium point, corresponding to an asymptotic presence of all states, are formulated. The case of γ=1 with asymmetric interaction rate is studied as well. Possible complex behavior of solutions can reflect the possible complex performance of systems with asymmetric interactions — typical e.g. in Economy and some applications in Biology.

Stochastic Modeling of Bacterial Population Growth with Antimicrobial Resistance

In this paper we consider a stochastic model of bacterial population growth with antimicrobial resistance under the influence of random fluctuations. We analyze the model for the problem of persistence and extinction of bacterial cells. This analysis shows asymptotic extinction and conditional persistence for growing population. Moreover, we perform computer simulations in order to illustrate the model behavior. The model results have important implications for the eradication of bacterial cells and the emergence of resistance.

GaiaData Release 3

Context.As part of the thirdGaiaData Release, we present the contributions of the non-stellar and classification modules from the eighth coordination unit (CU8) of the Data Processing and Analysis Consortium, which is responsible for the determination of source astrophysical parameters usingGaiadata. This is the third in a series of three papers describing the work done within CU8 for this release.Aims.For each of the five relevant modules from CU8, we summarise their objectives, the methods they employ, their performance, and the results they produce forGaiaDR3. We further advise how to use these data products and highlight some limitations.Methods.The Discrete Source Classifier (DSC) module provides classification probabilities associated with five types of sources: quasars, galaxies, stars, white dwarfs, and physical binary stars. A subset of these sources are processed by the Outlier Analysis (OA) module, which performs an unsupervised clustering analysis, and then associates labels with the clusters to complement the DSC classification. The Quasi Stellar Object Classifier (QSOC) and the Unresolved Galaxy Classifier (UGC) determine the redshifts of the sources classified as quasar and galaxy by the DSC module. Finally, the Total Galactic Extinction (TGE) module uses the extinctions of individual stars determined by another CU8 module to determine the asymptotic extinction along all lines of sight for Galactic latitudes |b|> 5°.Results.GaiaDR3 includes 1591 million sources with DSC classifications; 56 million sources to which the OA clustering is applied; 1.4 million sources with redshift estimates from UGC; 6.4 million sources with QSOC redshift; and 3.1 million level 9 HEALPixes of size 0.013 deg2where the extinction is evaluated by TGE.Conclusions.Validation shows that results are in good agreement with values from external catalogues; for example 90% of the QSOC redshifts have absolute error lower than 0.1 for sources with empty warning flags, while UGC redshifts have a mean error of 0.008 ± 0.037 if evaluated on a clean set of spectra. An internal validation of the OA results further shows that 30 million sources are located in high confidence regions of the clustering map.

Asymptotic extinction and persistence of a perturbed epidemic model with different intervention measures and standard Lévy jumps

Controlling an outbreak through response measures is critical to saving lives and protecting vulnerable populations. This article proposes an epidemic model with three intervention measures: media coverage, isolation, and medical therapy. Since randomness plays an important role in biology, from the molecular level to the organismal level, we extend our system to a more realistic framework, which then takes into account the effect of standard jumps due to some sudden environmental changes. After providing the associated framework, the sharp criteria for asymptotic extinction and persistence of illness are derived. To check the accuracy of our results, we perform two numerical examples.

Identification of transmission rates and reproduction number in a SARS-CoV-2 epidemic model

<p style='text-indent:20px;'>We consider a compartment mathematical model for a SARS-CoV-2 epidemic involving five classes considered to be essential to depict the feature of the epidemic. A first goal is to approach by an optimal control technique the identification of two essential transmission rates in the model, that is the average number of individuals infected in unit time by an infected symptomatic and by an asymptomatic, respectively. These are provided by the first-order conditions of optimality corresponding to the minimization problem introduced for formulating the identification objective. The discussion of the asymptotic stability of the system done for the case when life immunity is gained reveals an asymptotic extinction of the disease, with a well determined reproduction number.</p>

Persistence and extinction for stochastic ecological models with internal and external variables.

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research.

Bifurcations analysis of oscillating hypercycles

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n>4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named QSS). It has also been reported that, for values below QSS, the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q, undergoes a critical fidelity threshold, QPO. The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q<QPO. Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of QPO is slightly higher than QSS, thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q=1.

Pushed beyond the brink: Allee effects, environmental stochasticity, and extinction

To understand the interplay between environmental stochasticity and Allee effects, we analyse persistence, asymptotic extinction, and conditional persistence for stochastic difference equations. Our analysis reveals that persistence requires that the geometric mean of fitness at low densities is greater than one. When this geometric mean is less than one, asymptotic extinction occurs with high probability for low initial population densities. Additionally, if the population only experiences positive density-dependent feedbacks, conditional persistence occurs provided the geometric mean of fitness at high population densities is greater than one. However, if the population experiences both positive and negative density-dependent feedbacks, conditional persistence only occurs if environmental fluctuations are sufficiently small. We illustrate counter-intuitively that environmental fluctuations can increase the probability of persistence when populations are initially at low densities, and can cause asymptotic extinction of populations experiencing intermediate predation rates despite conditional persistence occurring at higher predation rates.

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population.

Stand dynamics and tree coexistence in an analytical structured model: The role of recruitment

Understanding the mechanisms of coexistence and niche partitioning in plant communities is a central question in ecology. Current theories of forest dynamics range between the so-called neutral theories which assume functional equivalence among coexisting species to forest simulators that explain species assemblages as the result of tradeoffs in species individual strategies at several ontogenetic stages. Progress in these questions has been hindered by the inherent difficulties of developing analytical size-structured models of stand dynamics. This precludes examination of the relative importance of each mechanism on tree coexistence. In previous simulation and analytical studies emphasis has been given to interspecific differences at the sapling stage, and less so to interspecific variation in seedling recruitment. In this study we develop a partial differential equation model of stand dynamics in which competition takes place at the recruitment stage. Species differ in their size-dependent growth rates and constant mortality rates. Recruitment is described as proportional to the basal area of conspecifics, to account for fecundity and seed supply per unit of basal area, and is corrected with a decreasing function of species specific basal area to account for competition. We first analyze conditions for population persistence in monospecific stands and second we investigate conditions of coexistence for two species. In the monospecific case we found a stationary stand structure based on an inequality between mortality rate and seed supply. In turn, intra-specific competition does not play any role on the asymptotic extinction or population persistence. In the two-species case we found that coexistence can be attained when the reciprocal negative effect on recruitment follows a given relation with respect to intraspecific competition. Specifically a tradeoff between recruitment potential (i.e. shade tolerance or predation avoidance) and fecundity or growth rate. This is to our knowledge the first study that describes coexistence mechanisms in an analytical size-structured model in terms of competitive differences at the regeneration state.

Chronic dietary magnesium-L-threonate speeds extinction and reduces spontaneous recovery of a conditioned taste aversion

Elevation of brain magnesium enhances synaptic plasticity and extinction of conditioned fear memories. This experiment examined the generalizability of this phenomenon by studying the effects of a novel magnesium compound, magnesium-L-threonate (MgT), on conditioned taste aversion (CTA) extinction and spontaneous recovery (SR). Adult male Sprague–Dawley rats were maintained on a 23-hour water deprivation cycle and acquired a CTA following the taste of a CS [0.3% saccharin+16mg/ml MgT (SAC+MgT)] paired with a US [81mg/kg (i.p.) lithium chloride (LiCl)]. Following CTA acquisition, rats drank a water+MgT solution for up to 1hour/day over the next 31days. For 14 additional days, some animals continued water+MgT treatment, but others drank water only to allow MgT to be eliminated from the body. We then employed 2 different extinction paradigms: (1) CS-Only (CSO), in which SAC was presented, every-other day, or (2) Explicitly Unpaired (EU), in which both SAC and LiCl were presented, but on alternate days. EU extinction procedures have been shown to speed CTA extinction and reduce spontaneous recovery of the aversion. Throughout extinction, half of the rats in each group continued to drink MgT (now in SAC or supplemental water+MgT solution), whereas the other half drank SAC only/water only until SAC drinking reached ≥90% of baseline (asymptotic extinction). Rats receiving MgT just before/during extinction drank less SAC on the first day of extinction suggesting that they had retained a stronger CTA. MgT enhanced the rate of extinction. Furthermore, the MgT-treated rats showed a relatively modest SR of the CTA 30days later — indicating that the extinction procedure was more effective for these animals. Our data suggest that long-term dietary MgT may enhance the consolidation/retention of a CTA, speed extinction, and inhibit SR of this learned aversion.

Stimulation of the dorsal periaqueductal gray enhances spontaneous recovery of a conditioned taste aversion

Due to its relevance to clinical practice, extinction of learned fears has been a major focus of recent research. However, less is known about the means by which conditioned fears re-emerge (i.e., spontaneously recover) as time passes or contexts change following extinction. The periaqueductal gray represents the final common pathway mediating defensive reactions to fear and we have reported previously that the dorsolateral PAG (dlPAG) exhibits a small but reliable increase in neural activity (as measured by c-fos protein immunoreactivity) when spontaneous recovery (SR) of a conditioned taste aversion (CTA) is reduced. Here we extend these correlational studies to determine if inducing dlPAG c-fos expression through electrical brain stimulation could cause a reduction in SR of a CTA. Male Sprague-Dawley rats acquired a strong aversion to saccharin (conditioned stimulus; CS) and then underwent CTA extinction through multiple non-reinforced exposures to the CS. Following a 30-day latency period after asymptotic extinction was achieved; rats either received stimulation of the dorsal PAG (dPAG) or stimulation of closely adjacent structures. Sixty minutes following the stimulation, rats were again presented with the saccharin solution as we tested for SR of the CTA. The brain stimulation evoked c-fos expression around the tip of the electrodes. However, stimulation of the dPAG failed to reduce SR of the previously extinguished CTA. In fact, dPAG stimulation caused rats to significantly suppress their saccharin drinking (relative to controls) – indicating an enhanced SR. These data refute a cause-and-effect relationship between enhanced dPAG c-fos expression and a reduction in SR. However, they highlight a role for the dPAG in modulating SR of extinguished CTAs.

Periaqueductal gray c-Fos expression varies relative to the method of conditioned taste aversion extinction employed

A conditioned taste aversion (CTA) is acquired when an animal consumes a novel taste (CS) and then experiences the symptoms of poisoning (US). Following CTA training, animals will avoid the taste that was previously associated with malaise. This defensive reaction to a learned fear can be extinguished by repeated exposure to the CS alone (CS-only; CSO-EXT). However, following a latency period in which the CS is not presented, the CTA will spontaneously recover (SR). Through the use of an explicitly unpaired extinction procedure (EU-EXT) we have shown that we can speed up extinction and attenuate SR of the CTA. Here we compared and contrasted the ability of CSO and EU extinction procedures to affect c-Fos expression in the periaqueductal gray (PAG). Fluid-deprived Sprague–Dawley rats acquired a strong CTA [via 3 pairings of 0.3% oral saccharin (SAC; the CS) and 81 mg/kg i.p. lithium chloride (LiCl; the US)] followed by extinction trials consisting of multiple exposures to either, (a) the CS every-other day (CSO-EXT), or (b) CS and US on alternate days (EU-EXT). A different group of rats did not receive multiple CS exposures and served as a “no extinction” (NE) control. Both extinction procedures resulted in ≥ 90% reacceptance of SAC (achieving asymptotic extinction). Some of the animals were sacrificed for c-Fos immunohistochemical analysis following asymptotic extinction. Other rats entered a 30-day latency period where they drank water only. These remaining animals were then tested for SR with a final exposure to SAC before being sacrificed for c-Fos immunohistochemistry. As reported previously, rats in the CS-only group exhibited a significant SR of the CTA. However, animals in the EU extinction group reached asymptotic extinction more rapidly than did CSO rats and they did not show SR of the CTA. As compared to rats that retained their CTA, both groups of extinguished rats showed suppression in the number of c-Fos-labeled neurons in all 4 longitudinal columns of the PAG. The number of c-Fos-labeled cells in the PAG was generally low but there was a reliable increase in c-Fos expression in dorsolateral PAG (dlPAG) following the SR test in the brains of rats that went through the EU-EXT procedure as compared with those that either went through the more-traditional CSO extinction procedure or experienced no extinction at all. The number of c-Fos-labeled neurons in the dlPAG was significantly correlated with the amount of SAC consumed at the SR test. Surprisingly, the brains of EU-extinguished rats and CSO extinguished rats did not differ in the number of c-Fos-labeled neurons in gustatory neocortex, medial prefrontal cortex, basolateral amygdala, or the central nucleus of the amygdala. Thus, behavioral differences in SR between the EU and CSO extinction animals were not represented by corresponding changes in the neural activity of several brain nuclei classically associated with extinction learning. However a detailed analysis of PAG c-Fos expression provided hints about some of the physiological changes evoked by these 2 extinction paradigms that produce very different behavioral outcomes. The findings are clinically relevant as we seek the development of treatments for deficits in fear extinction (e.g. PTSD, phobias).

Acute, but not chronic, exposure to d-cycloserine facilitates extinction and modulates spontaneous recovery of a conditioned taste aversion

d-cycloserine, the glutamate N-methyl-d-aspartate receptor partial agonist, has been reported to facilitate the extinction of learned fears acquired in both naturalistic and laboratory settings. The current study extended this literature by evaluating the ability of either chronic or acute administrations of DCS to modulate the extinction and spontaneous recovery of a conditioned taste aversion (CTA).Twenty-three hour fluid-deprived Sprague–Dawley rats acquired a strong CTA following 3 pairings of a conditioned stimulus (CS; 0.3% oral saccharin)+unconditioned stimulus [US; 81mg/kg (i.p.) lithium chloride (LiCl)]. In separate groups of rats, we then employed 2 different extinction paradigms: (1) CS-only (CSO-EXT) in which saccharin was presented every-other day, or (2) Explicitly Unpaired (EU-EXT) in which both saccharin and LiCl were presented but on alternate days. Previous studies have indicated that the EU-EXT procedure speeds up the extinction process. Further, spontaneous recovery of a CTA emerges following CSO-EXT but the EU-EXT paradigm causes a suppression of spontaneous recovery. DCS (15mg/kg, i.p.) was administered immediately after daily liquid presentations (saccharin or water, alternate days) during the extinction period. In an acute drug manipulation, DCS (15mg/kg, i.p.) or saline control injections were administered for 4days only. This was done during one of 3 different phases of extinction [i.e., static (2–5%), early dynamic (8–16%), or middle dynamic (20–40%) saccharin reacceptance]. Other animals assigned to the chronic DCS condition received daily DCS (15mg/kg, i.p.) throughout extinction. Changes in saccharin drinking in these animals were compared to the data from rats that received no drug (saline controls). Once rats met our criterion for asymptotic extinction (90% reacceptance of the CS) they entered a 30-day latency period during which they received water for 1h/day. The day after the completion of the latency period, a final opportunity to drink saccharin was provided (spontaneous recovery test).Saline-treated control rats that went through the EU-EXT procedure achieved asymptotic extinction more quickly than did the CSO-EXT rats and did not exhibit a spontaneous recovery of the CTA. Chronic DCS treatments did not significantly reduce the time to achieve asymptotic CTA extinction in rats exposed to either CSO or EU extinction methods. Further, animals treated with DCS throughout EU-EXT exhibited a spontaneous recovery of the CTA whereas the saline-treated, EU-EXT rats did not. Thus, chronic DCS treatment did not shorten the time to extinguish a CTA and this treatment eliminated the ability of EU-EXT to block spontaneous recovery of the CTA. Acute DCS treatments were more effective in reducing the time required to extinguish a CTA than were chronic drug treatments. Moreover, the timing of these acute DCS treatments affected spontaneous recovery of the CTA depending on the extinction method employed. Acute DCS administrations later in extinction were more effective in reducing spontaneous recovery than were early administrations if the rats went through the CSO-EXT procedure. However, late-in-extinction administrations of DCS facilitated spontaneous recovery of the CTA in rats that experienced the EU-EXT method.These data agree with other findings suggesting that DCS treatments are more effective when administered a limited number of times. Our data extend these findings to the CTA paradigm and further suggest that, depending on the extinction paradigm employed, acute exposure to DCS can speed up CTA extinction and reduce spontaneous recovery of the aversion. The timing of the acute DCS treatment during extinction is generally less important than its duration in predicting the rate of CTA extinction. However, the timing of acute DCS treatments during extinction and the method of extinction employed can interact to affect spontaneous recovery of a CTA.

A new explanation of the extinction paradox

This work presents a new explanation for the extinction paradox and shows that the canonical explanations are incorrect. This paradox refers to the large size limit of a particle's extinction cross section. It is called a paradox because the geometrical optics approximation, which should be valid in this limit, predicts a cross section that is half of the true value. The new explanation is achieved by formulating the scattered wave in terms of an integral over the particle's surface where the seemingly unrelated Ewald–Oseen theorem appears in the formulation. By expressing the cross section in terms of this surface integral, the Ewald–Oseen theorem is analytically connected to the cross section. Several illustrations are used to reveal the significance of this connection: The paradox is seen to be a consequence of the requirement that the incident wave be canceled within the particle by secondary radiation from its own internal field. Following this, the canonical explanations are examined to reveal serious problems. In the process, the same asymptotic extinction behavior is shown to occur for small highly refractive dielectric particles, and thus is not just a large particle size or small wavelength effect as is often stated. The traditional explanations cannot account for this behavior while the new one actually predicts it. All in all, this work constitutes a fundamental reworking of 60 years of accepted understanding for the cause of the asymptotic behavior of the extinction cross section.

Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles

The extinction efficiency factor associated with the scattering of a plane electromagnetic wave impinging on a basal face of a dielectric disk or a cylindrical particle is investigated by employing the physical-geometric optics hybrid (PGOH) method and the discrete-dipole approximation (DDA) method. It is found that the derived extinction efficiency factor from the PGOH is a function of the thickness of the disk, or the length of the cylinder, and the refractive index, but is independent of the diameter and shape of the cross section of the basal face of the particle. Furthermore, the oscillations of the extinction efficiency factor versus the thickness or length of the particle do not diminish if the particle is not absorptive. The values of the extinction efficiency factor simulated from the DDA method are quite different from those computed from the PGOH, although the size parameter of the particle is in the commonly recognized geometric optics regime. To explain the difference, the concept of the edge effect associated with the tunneling rays in the semiclassical scattering theory is generalized from the case of spherical particles to that of nonspherical particles based on the localization principle. Accordingly, the edge-effect contribution can be distinguished and removed from the extinction cross section calculation by the DDA method. The remaining part of the extinction cross section, associated with the interference between the transmitted rays and incident rays, agrees well with the results computed from the PGOH, and the agreement illustrates the presence of the edge effect in the case of nonspherical particles with surfaces that have no curvature along the incident direction. It is found that the asymptotic extinction efficiency factor may not necessarily converge to 2, but it depends on the specific physical processes of the interference between diffracted and transmitted light and of the edge effect.

Explicit disassociation of a conditioned stimulus and unconditioned stimulus during extinction training reduces both time to asymptotic extinction and spontaneous recovery of a conditioned taste aversion

Conditioned taste aversions (CTAs) may be acquired when an animal consumes a novel taste (CS) and then experiences the symptoms of poisoning (US). This aversion may be extinguished by repeated exposure to the CS alone. However, following a latency period in which the CS is not presented, the CTA will spontaneously recover (SR). In the current study we employed an explicitly unpaired extinction procedure (EU-EXT) to determine if it could thwart SR of a CTA. Sprague–Dawley rats acquired a strong CTA after three pairings of saccharin (SAC the CS) and Lithium Chloride (LiCl the US). CTA acquisition was followed by extinction (EXT) training consisting of either (a) CS-only exposure (CSO) or (b) exposure to saccharin and Lithium Chloride on alternate days (i.e., explicitly unpaired: EU). Both extinction procedures resulted in ⩾90% reacceptance of SAC, although the EU extinction procedure (EU-EXT) significantly decreased the time necessary for rats to reach this criterion (compared to CSO controls). Rats were subsequently tested for SR of the CTA upon re-exposure to SAC following a 30-day latency period of water drinking. Rats that acquired a CTA and then underwent the CSO extinction procedure exhibited a significant suppression of SAC drinking during the SR test (as compared to their SAC drinking at the end of extinction). However, animals in the EU-EXT group did not show such suppression in drinking compared to CSO controls. These data suggest that the EU-EXT procedure may be useful in reducing both time to extinction and the spontaneous recovery of fears.

Properties of super-Poisson processes and super-random walks with spatially dependent branching rates

The global supports of super-Poisson processes and super-random walks with a branching mechanism ϕ(z) = z 2 and constant branching rate are known to be noncompact. It turns out that, for any spatially dependent branching rate, this property remains true. However, the asymptotic extinction property for these two kinds of superprocesses depends on the decay rate of the branching-rate function at infinity.

On Interfaces and Oscillatory Solutions of Higher-Order Semilinear Parabolic Equations with Non-Lipschitz Nonlinearities

We study local properties of solutions and their asymptotic extinction behavior for the fourth-order semilinear parabolic equation of diffusion-absorption type where p < 1, so that the absorption term is not Lipschitz continuous at u = 0. The Cauchy problem with bounded compactly supported initial data possesses solutions with finite interfaces, and we describe their oscillatory, sign changing properties for p ∈ (-1 3 1). For p ∈ (0, 1), we also study positive solutions of the free-boundary problem with zero contact angle and zero-flux conditions. Finally, we describe families {f k } of similarity extinction patterns u s (x, t) = (T - t) 1/(1-p) f(y), where y = x/(T- t) 1/4 , that vanish in finite time, as t → T - ∈ (0, ∞). Similar local and asymptotic properties are indicated for the sixth-order equation with source u t = u xxxxxx ± |u| p-1 u , where p ∈ (-1 5, 1).

Comparison results and steady states for the Fujita equation with fractional Laplacian

We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman–Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions. By extending the method of moving planes to fractional powers of the Laplacian we prove that all positive steady states of the corresponding equation in a finite ball are radially symmetric.