Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.     

Stability of Spherically Symmetric Steady States in Galactic Dynamics¶against General Perturbations

Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a non-linear stability property of such steady states. In previous investigations by Y. Guo and G. Rein, stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this non-physical restriction.     

Effect of Coulomb Damping on Buckling of a Two-Rod System

This paper describes a significant influence of a slight Coulomb damping on buckling, using a simple two rods system. Coulomb damping produces equilibrium regions around the well-known stable and unstable steady states under the pitchfork bifurcation which occurs in the case without Coulomb damping. Also, the stability of the states in the equilibrium regions is examined by using the phase portrait. As a consequence, due to the slight Coulomb damping, it is theoretically clarified that the states in the equilibrium regions are locally stable, even in the neighborhood of the unstable steady states under the pitchfork bifurcation in the case without Coulomb damping, i.e., even in the neighborhood of the unstable trivial steady states in the postbuckling and the unstable nontrivial steady states under the subcritical pitchfork bifurcation. Furthermore, the experimental results are in qualitative agreement with the theoretically predicted phenomena.     

Stability analysis of steady rotations of a rigid body bearing two-degree-of-freedom control moment gyros with dissipation in gimbal suspension axes

To study the stability of steady rotations of a control moment gyro system with internal dissipation, we use the Barbashin-Krasovskii theorem and the relation, established in [1], between the Lyapunov function and steady motions. Taking into account the special properties of the original problem, we reduce it to a lower-dimensional problem.We give a detailed presentation of an algorithm for analyzing the stability of steady motions of a gyrostat and use this algorithm to perform a complete study for two systems consisting, respectively, of one and two gyros whose gimbal axes are parallel to the principal axis of inertia of the system. Each steady motion is identified as either asymptotically stable or unstable. We find periodic motions that exist only in the presence of dynamic symmetry and which are regular precessions. For the system with two gyros, we prove the asymptotic stability of quiescent states and prove that in the angular momentum range where these states are defined the system does not have any other stable motions.     

A review of rotating cylinder wake transitions

Recent work on the flow past a rotating cylinder is reviewed and further investigated at low Reynolds numbers. The various two- and three-dimensional transitions that occur as the rotation rate is increased are detailed. Two steady states, steady state I and steady state II, are identified based on the physical characteristics of the wake and the drag force on the body. Steady state I occurs at lower rotation rates, while state steady state II occurs at higher rotation rates. Linear stability analysis shows that two three-dimensional modes become unstable on steady state I and steady state II. Floquet stability analysis of the unsteady base flows that occur at very low rotation rates shows the presence of five three-dimensional modes. The curves of marginal stability are presented, followed by a comparison of numerical simulations to their experimentally obtained counterparts. Furthermore, the spatio-temporal characteristics of each mode and the likely underlying physical mechanisms are briefly discussed.     

A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. I Equal diffusivities

A simple model for homogeneous-heterogeneous reactions in stagnation-point boundary-layer flow is constructed in which the homogeneous (bulk) reaction is assumed to be given by isothermal cubic autocatalator kinetics and the heterogeneous (surface) reaction by first order kinetics. The possible steady states of this system are analysed in detail in the case when the diffusion coefficients of both reactant and autocatalyst are equal. Hysteresis bifurcations leading to multiple solutions are found. The temporal stability of these steady states is then discussed.     

Uniqueness and stability of steady states of continuous chemical reactors

We examine the uniqueness and stability of the solutions to the problem of steady-state operation of a continuous chemical reactor in which longitudinal diffusion and heat conduction are taken into account. We investigate an adiabatic reactor in which the concentration and temperature distributions are similar (the thermal diffusivity and diffusion coeffecient are equal) and an isothermic reactor. These two cases are considered together because the mathematical formulations of the problem are equivalent.The question of the existence and number of steady states was analyzed in [1, 2], where references were made to earlier investigations. The results obtained in [1, 2] are now extended. The stability of the steady states is investigated by the small-perturbation method.     

Bifurcation analysis of a diffusive predator–prey system with nonmonotonic functional response

Nonlinear Dynamics - In this paper, a diffusive predator–prey model with nonmonotonic functional response is investigated. The stability of the positive spatially homogeneous steady states...     

Global properties of a class of virus infection models with multitarget cells

In this paper, we propose a class of virus infection models with multitarget cells and study their global properties. We first study three models with specific forms of incidence rate function, then study a model with a more general nonlinear incidence rate. The basic model is a (2n+1)-dimensional nonlinear ODEs that describes the population dynamics of the virus, n classes of uninfected target cells, and n classes of infected target cells. Model with exposed state and model with saturated infection rate are also studied. For these models, Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models. We have proven that if the basic reproduction number is less than unity then the uninfected steady state is globally asymptotically stable, and if the basic reproduction number is greater than unity then the infected steady state is globally asymptotically stable. For the model with general nonlinear incidence rate, we construct suitable Lyapunov functions and establish the sufficient conditions for the global stability of the uninfected and infected steady states of this model.     

Non-Symmetric Convective Flows in Laterally Heated Rectangular Cavities

Bifurcations of central symmetry breaking and stability of nonsymmetric states of buoyancy-driven convection in laterally heated cavities are studied numerically. The calculations are carried out using two independent numerical approaches. Stability and weakly nonlinear analysis of the calculated bifurcations are studied by the spectral Galerkin method. Time-marching calculations are carried out using the finite volume method. By applying two independent numerical approaches the subcritical steady flows, their stability, the transitions between different states and flows at small and large supercriticalities are comprehensively investigated. It is shown how these numerical techniques can be applied for interpreting a particular experimental result.     

Convection in a low Prandtl number fluid layer rotating about a vertical axis

Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found.     

BIFURCATION ANALYSIS OF A MITOTIC MODEL OF FROG EGGS

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Wave features of a hyperbolic reaction–diffusion model for Chemotaxis

The Extended Thermodynamic theory is used to derive a hyperbolic reaction–diffusion model for Chemotaxis. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. A particular emphasis is given to the occurrence of the Turing bifurcation. The existence of traveling wave solutions connecting the two steady states is investigated and the governing equations are numerically integrated to validate the analytical results. The propagation of plane harmonic waves is analyzed and the stability regions in terms of the model parameters are shown. The frequency dependence of the phase velocity and of the attenuation is also illustrated. Finally, in order to have a measure of the non linear stability, the propagation of acceleration waves is studied, the wave amplitude is derived and the critical time is evaluated.     

On the Resonant Generation of Weakly Nonlinear Stokes Waves in Regions with Fast Varying Topography and Free Surface Current

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｉｎｔｅｒａｃｔｉｏｎｏｆｓｕｒｆａｃｅｗａｔｅｒｗａｖｅｓｗｉｔｈａｍｂｉｅｎｔｃｕｒｒｅｎｔｓａｎｄｕｎｄｕｌａｔｉｎｇｓｅａｂｅｄｔｏｐｏｇｒａｐｈｙｉｓｏｆｆｕｎｄａｍｅｎｔａｌｉｍｐｏｒｔａｎｃｅｔｏｃｏａｓｔａｌｅｎｇｉｎｅｅｒｓａｎｄｓｅｄｉｍｅｎｔｏｌｏｇｉｓｔｓ.Ｆｏｒｅｘａｍｐｌｅ,ｔｈｅｒｅｓｏｎａｎｔｇｅｎｅｒａｔｉｏｎｏｆｓｕｒｆａｃｅｗａｖｅｓｉｎａｃｕｒｒｅｎｔｏｖｅｒｏｔｈｅｒｔｉｄａｌｌｙｏｒｗａｖｅｉｎｄｕｃｅｄｂｅｄｆｏｒｍｓ,ｓ…     

破碎岩体非等温渗流的非线性动力学研究

分别从固体及流体导热的能量方程出发,导出破碎岩体非等温渗流的能量本构方程, 结合渗流的连续性方程、运动方程、状态方程等建立了破碎岩体非等温渗流的一维非线性动力学方程组;结合Mathcad软件计算得到了系统的无量纲化平衡态, 利用逐次亚松弛迭代法分析了对应于不同参数时平衡态的稳定性;指出非等温渗流系统存在鞍结分岔及折叠突变, 与等温渗流相比, 考虑温度场的破碎岩体渗流动力系统更容易发生渗流突变.      

The torsional stability of a cylinder subject to finite perturbations

The problem of torsional stability of a circular cylinder made from a compressible nonlinearly elastic material is solved for finite perturbations. In contrast to the classical theory of bifurcation, an infinite sequence of steady states that bounds the domain of allowed initial perturbations is constructed. The applicability of the classical three-dimensional linear theory of stability is evaluated. Voronezh University, Russia. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 133–136, March, 2000.     

Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations

We study the linear stability of smooth steady states of the evolution equation

Effect of delay on a predator–prey model with parasitic infection

In the paper an eco-epidemic system with delay and parasitic infection in the prey is investigated. The conditions for asymptotic stability of steady states are derived and the length of the delay preserving the stability is also estimated. Further, the criterion for existence of Hopf-type small amplitude periodic oscillations of the predator and prey biomass is derived. Numerical results indicate that the delay does not affect the stability of the system in the process but makes all populations oscillate more intensively. In addition, the results show that the recovery makes the levels of the infected prey and the predator become lower but makes the sound prey higher in limit time.     

On the Structure and Stability of the Set of Solutions of a Nonlocal Problem Modeling Ohmic Heating

In this work we analyze the existence, structure and stability of the set of positive solutions of a nonlocal elliptic boundary value problem modeling Ohmic heating. By regarding the total electric current flowing through the device by unit of cross-sectional area as the continuation parameter, we show that the set of steady states is constituted by a finite number of differentiable curves. Moreover, the instability index of a steady state changes by 1 when we pass through by a turning point and does not change if we pass through an hysteresis point. Some sufficient conditions so that the model admit a positive steady state for each value of the parameter, as well as for uniqueness and non-existence, will be also given.