Transcritical Point Research Articles

Dynamics of a Class of Prey–Predator Models with Singular Perturbation and Distributed Delay

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter [Formula: see text], the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.

THE EFFECT OF AN ADDITIVE NOISE ON SOME SLOW-FAST EQUATION NEAR A TRANSCRITICAL POINT

We consider the effect of small additive noise with intensity $\sigma$ on trajectories of a slow-fast system with small parameter $\varepsilon$ which admits bifurcation delay at a transcritical point. We estimate the probability that the perturbed stochastic paths stay in some tubular neighborhood of the deterministic path to show that small but not exponentially small noise destroys the bifurcation delay caused by transcritical point and obtain a noise intensity threshold value $N(\varepsilon)$ of order $\varepsilon^{\frac{3}{4}}$. When <i>e</i>^{-1/$\varepsilon$}《<i>σ</i>＜$N(\varepsilon)$, the paths are likely to leave the neighborhood of the corresponding determinate path before some time of order $\sqrt{\varepsilon|log\sigma|}$. When $\sigma>N(\varepsilon) $, the paths are likely to leave before some time of order $\sigma^{\frac{2}{3}}$.

Application Progress of Carbon Dioxide in Chemical Industry by Molecular Simulation Technology

CO₂ has attracted much attention due to its unique physical and chemical properties. Supercritical CO₂ has been widely used because of its mild conditions, non-toxicity and wide sources. In the field of material chemistry, molecular simulation technology, as a new scientific research method, plays an important role in the basic theory and application of CO₂. This paper reviews the progress of molecular simulation technology in the field of CO₂research in recent years, including physical properties, supercritical point properties, transcritical point characteristics, diffusion, adsorption separation, extraction and other mass transfer properties, as well as interface properties. Compared with the traditional method, the computer simulation method has obvious advantages.

Effect of predators' behavior on prey-predator interaction

In this paper, we study the effect of the behavior of predators on prey-predator interaction. We assume that there is one prey population that suffers from the fear of predators which could force it to move in to the refuge; and three are two predator populations, one of them is aggressive in its attack and the other one using sit-and-wait procedure, which is less aggressive. All the model’s equilibrium points have been found and their stability was established. The possibility of transcritical and Hopf bifurcation was also investigated and numerical simulations were given. The effect of prey refuge and fear also are detected. The cost of them is to allow the model to reach to double transcritical point. The effect of the competition between prey populations is to convert the model from the stable limit cycle to a spiral stable equilibrium point of afraid prey with predator. When it becomes large it converts the model to the stable trivial solution.

Thermocapillary instability on a film falling down a non-uniformly heated slippery incline

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number Mn as well as slip length δ. This depicts that δ and Mn both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of Mn and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of Mn, when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude (ζa) decreases (increases) with the increment of Mn and δ. Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum hmax and minimum hmin thickness amplifies, for small change of Mn and δ. Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ. The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as Mn increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ>max16Mn−13,12Mn−23−Mn.

Impact of delay on HIV-1 dynamics of fighting a virus with another virus.

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number R0 are identified and its threshold properties are discussed. When R0<1, the infection-free equilibrium E0 is globally asymptotically stable. When R0>1, E0 becomes unstable and there occurs the single-infection equilibrium Es, and E0 and Es exchange their stability at the transcritical point R0=1. If 1<R0<R1, where R1 is a positive constant explicitly depending on the model parameters, Es is globally asymptotically stable, while when R0>R1, Es loses its stability to the double-infection equilibrium Ed. There exist a constant R2 such that Ed is asymptotically stable if R1<R0<R2, and Es and Ed exchange their stability at the transcritical point R0=R1. We use one numerical example to determine the largest range of R0 for the local stability of Ed and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling.

Note from the Hubbert Quorum

Department of Earth and Planetary Sciences, University of California-Santa Cruz, Santa Cruz, CA, USASince 1993 the ‘‘Hubbert Quorum’’ has met annually onthe day prior to the Fall Meeting of the American Geophysi-cal Union (AGU) in San Francisco. The purpose of this one-day meeting at the USGS-Menlo Park facility is to bringtogether hydrogeologists and geoscientists who are inter-ested in geologic processes associated with crustal fluids.Frequently these researchers work on topics that haveapplication to fields outside of hydrology such as faultmechanics, volcanology, and economic geology (see http://wwwrcamnl.wr.usgs.gov/hydrotherm/Hubbert.htm).The Hubbert Quorum was originally inspired by thesuccess of the Gilbert Club, which has met annually since1983 on the day after the AGU Fall Meeting (see http://socrates.berkeley.edu/~geomorph/gilbert.htm). The Gil-bert Club, which takes place at the University of Califor-nia-Berkeley, was conceived as an opportunity for thegeomorphology community to hear major talks, debatenew ideas, and encourage community-level discussion ofissues. It is named after the great geoscientist Grove KarlGilbert (Bourgeois, 1998). Similarly, the Hubbert Quoruminvokes the name of the eminent geophysicist Marion KingHubbert, who contributed immensely to the understand-ing of the role of fluids in geologic processes (e.g., Hub-bert, 1940, 1953; Hubbert and Willis, 1957; Hubbert andRubey, 1959) and also had an abiding interest in the sig-nificance of Earth resources in human affairs (e.g., Hub-bert, 1949, 1956, 1967).Like the Gilbert Club, the Hubbert Quorum has no for-mal affiliation with any organization and has always beenan open meeting, advertised through email. The intent isto enable an informal group of natural-process-orientedscientists to freely exchange ideas on an annual basis. Pre-sentations are made on a voluntary basis and all partici-pants are invited to present their work.The Quorum is an all-day affair beginning at 11:00 AMand adjourning at 6:00 PM. The meeting begins with an‘‘icebreaker’’ brunch and is followed by several extended(20 min), and short (10 min) presentations. The goal is toavoid the hectic pace of typical meetings by providingample time for open discussion and questions during andfollowing each presentation. All speakers are encouragedto initiate discussion by presenting work in progress, ratherthan a canned talk.The past few decades have seen substantial growth ofinterest in the role of fluids in geologic processes. Hydro-geologists and other geoscientists are actively exploringthe role of groundwater and other subsurface fluids insuch fundamental geologic processes as crustal heat trans-fer, ore deposition, hydrocarbon migration, earthquakes,tectonic deformation, diagenesis, and metamorphism. Thegrowing interest in such problems is indicated by thepopularity of regularly scheduled meetings on ‘‘Water-Rock Interaction’’ and ‘‘Geofluids,’’ by Penrose Confer-ences such as ‘‘Continental Magmatic-HydrothermalSystems’’ (2004), and by an increasing number of relatedsymposia at annual meetings of organizations such asAGU.In keeping with this community-wide trend, the 15thHubbert Quorum on December 9, 2007, was the largestever, attracting 55 participants who represented a wide rangeof topical expertise and diverse institutions and countries.The first two speakers were from ETH-Zurich and presentedcomplementary talks that described progress in modelingmultiphase, multicomponent, higher-temperature (trans-critical point) hydrothermal flow. Thomas Driesnerdescribed some of the associated computational challengesin his talk titled The not-so-trivial side of combining the ther-modynamics and flow of H

A bifurcation study of mixed-convection heat transfer in horizontal ducts

The bifurcation structure of two-dimensional, pressure-driven flows through a horizontal, rectangular duet that is heated with a uniform flux in the axial direction and a uniform temperature around the periphery is examined. The solution structure of the flow in a square duct is determined for Grashof numbers (Gr) in the range of 0 to 106using an arclength continuation scheme. The structure is much more complicated than reported earlier by Nandakumar, Masliyah &amp; Law (1985). The primary branch with two limit points and a hysteresis behaviour between the two-and four-cell flow structure that was computed by Nandakumaret al.is confirmed. An additional symmetric solution branch, which is disconnected from the primary branch (or rather connected via an asymmetric solution branch), is found. This has a two-cell flow structure at one end, a four-cell flow structure at the other, and three limit points are located on the path. Two asymmetric solution branches emanating from symmetry-breaking bifurcation points are also found for a square duct. Thus a much richer solution structure is found with up to five solutions over certain ranges of Or. A determination of linear stability indicates that all two-dimensional solutions develop some form of unstable mode by the time Gr is increased to about 220000. In particular, the four-cell becomes unstable to asymmetric perturbations. The paths of the singular points are tracked with respect to variation in the aspect ratio using the fold-following algorithm. Transcritical points are found at aspect ratios of 1.408 and 1.456 respectively for Prandtl numbersPr= 0.73 and 5. Above these aspect ratios the four-cell solution is no longer on the primary branch. Some of the fold curves are connected in such a way as to form a tilted cusp. When the channel cross-section is tilted even slightly (1°) with respect to the gravity vector, the bifurcation points unfold and the two-cell solution evolves smoothly as the Grashof number is increased. The four-cell solutions then become genuinely disconnected from the primary branch. The uniqueness range in Grashof number increases with increasing tilt, decreasing aspect ratio and decreasing Prandtl number.

Flow through rotating rectangular ducts

The bifurcation structure of two-dimensional, pressure-driven flows through a rectangular duct that is rotating about an axis perpendicular to its own is examined at a fixed Ekman number (Ek=ν/b2Ω) of 0.01. The solution structure for flow through a square duct (aspect ratio γ=1) is determined for Rossby numbers (Ro=U/bΩ) in the range of 0–5 using a computational scheme based on the arclength continuation method. The structure is much more complicated than reported earlier by Kheshgi and Scriven [Phys. Fluids 28, 2968 (1985)]. The primary branch with two limit points in Rossby number and a hysteresis behavior between the two- and four-cell flow structure that was computed by Kheshgi and Scriven is confirmed. An additional symmetric solution branch, which is disconnected from the primary branch (or rather connected via an asymmetric solution branch), is found. This has a two-cell flow structure at one end, a four-cell flow structure at the other and three limit points are located on the path. Two asymmetric solution branches emanating from symmetry breaking bifurcation points are also found for a square duct. Thus even within a Rossby number range of 0–5 a much richer solutions structure is found with up to five solutions at Ro=5. An eigenvalue calculation indicates that all two-dimensional solutions develop some form of unstable mode by the time Ro is increased to 5.0. In particular, the four-cell solution becomes unstable to asymmetric perturbations as found in a related problem of flow through a curved duct. The paths of the singular points are tracked with respect to variation in the aspect ratio using the fold following algorithm. A transcritical point is found at an aspect ratio of 0.815 and below which the four-cell solution is no longer on the primary branch. When the channel cross section is tilted even slightly (1°) with respect to the axis of rotation, the bifurcation points unfold and the two-cell solution evolves smoothly as Rossby number is increased. The four-cell solutions then become genuinely disconnected from the primary branch. The uniqueness range in Rossby number increases with increasing tilt.