Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

Basin boundaries and bifurcations near convective instabilities: a case study

Using a scalar advection-reaction-diffusion equation with a cubic nonlinearity as a simple model problem, we investigate the effect of domain size on stability and bifurcations of steady states. We focus on two parameter regimes, namely, the regions where the steady state is convectively or absolutely unstable. In the convective-instability regime, the trivial stationary solution is asymptotically stable on any bounded domain but unstable on the real line. To measure the degree to which the trivial solution is stable, we estimate the distance of the trivial solution to the boundary of its basin of attraction: We show that this distance is exponentially small in the diameter of the domain for subcritical nonlinearities, while it is bounded away from zero uniformly in the domain size for supercritical nonlinearities. Lastly, at the onset of the absolute instability where the trivial steady state destabilizes on large bounded domains, we discuss bifurcations and amplitude scalings.     

A competition un-stirred chemostat model with virus in an aquatic system

A reaction–diffusion system of two bacteria species competing a single limiting nutrient with the consideration of virus infection is derived and analysed. Firstly, the well-posedness of the system, the existence of the trivial and semi-trivial steady states, and some prior estimations of the steady states are given. Secondly, a single species subsystem with virus is studied. The stability of the trivial and semi-trivial steady states and the uniform persistence of the subsystem are obtained. Further, taking the infective ability of virus as a bifurcation parameter, the global structure of the positive steady states and the effect of virus on the positive steady states are established via bifurcation theory and limiting arguments. It shows that the backward bifurcation may occur. Some sufficient conditions for the existence, uniqueness and stability of the positive steady state are also obtained. Finally, some sufficient conditions on the existence of the positive steady states for the full system are derived by using the fixed point index theory. Some results on persistence or extinction for the full system are also obtained.     

Qualitative analysis of a reaction–diffusion system

In this paper, we study the qualitative analysis of a reaction–diffusion system. The local stability of the trivial steady state, as well as the blow-up behavior of the solution are obtained.     

Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations

We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.     

The asymptotic analysis of an insect dispersal model on a growing domain

This paper is concerned about a reaction-diffusion equation on n-dimensional isotropically growing domain, which describes the insect dispersal. The model for growing domains is first derived, and the comparison principle is then presented. The asymptotic behavior of the solution to the reaction-diffusion problem is given by constructing upper and lower solutions. Our results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady state solution while it takes a negative effect on the asymptotic stability of the trivial solution. Numerical simulations are also performed to illustrate the analytical results.     

Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics

We investigate a system of nonlinear differential equations with distributed delays, arising from a model of hematopoietic stem cell dynamics. We state uniqueness of a global solution under a classical Lipschitz condition. Sufficient conditions for the global stability of the population are obtained, through the analysis of the asymptotic behavior of the trivial steady state and using a Lyapunov function. Finally, we give sufficient conditions for the unbounded proliferation of a given cell generation.     

A Hölling’s type II prey-predator model with stage structure and nonlocal delay

In this paper, a prey-predator model with reaction-diffusion is investigated under homogenous Neumann boundary condition. By taking food ingestion and species’ moving into account, model is further coupled with Hölling’s type II function response and nonlocal delay. Sufficient conditions for the global stability of three equilibria, i.e. positive, semi-trivial and trivial steady states are mainly derived by Lyapunov functional, respectively. Results show that intra-specific competition benefits the coexistence of prey and predator. Numerical simulations are performed to illustrate the analytical results.     

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell (HSC) populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov–Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describe the uncontrolled proliferation of HSC population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.     

On the stability of hematopoietic model with feedback control

We propose and analyze a mathematical model of the production and regulation of blood cell population in the bone marrow (hematopoiesis). This model includes the primitive hematopoietic stem cells (PHSC), the three lineages of their progenitors and the corresponding mature blood cells (red blood cells, white cells and platelets). The resulting mathematical model is a nonlinear system of differential equations with several delays corresponding to the cell cycle durations for each type of cells. We investigate the local asymptotic stability of the trivial steady state by analyzing the roots of the characteristic equation. We also prove by a Lyapunov function the global asymptotic stability of this steady state. This situation illustrates the extinction of the cell population in some pathological cases.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.     

Travelling waves in an infectious disease model with a fixed latent period and a spatio–temporal delay

This paper is concerned with the existence of travelling waves to an infectious disease model with a fixed latent period and a spatio–temporal delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this model is discussed. By constructing a pair of upper–lower solutions, we use the cross iteration method and the Schauder’s fixed point theorem to prove the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

BIFURCATION OF TWO-DIMENSIONAL KURAMOTO-SIVASHINSKY EQUATION

This paper deals with the steady state bifurcation of the K-S equation in two spatial dimensions with periodic boundary value condition and of zero mean. With the increase of parameter a, the steady state bifurcation behaviour can be very complicated. For convenience, only the cases a=2 and a=5 witl be discussed. The asymptotic expressions of the steady state solutions bifurcated from the trivial solution near a=2 and a=5 are given. And the stability of thenontriviat sotutions bifurcated from a=2 is studied. Of course, the cases a=n^2 m^2,n,m∈N(a≠2,5) can be similarly discussed by the same method which is used to discussing the cases a=2 and a= 5.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.     

Global stability and travelling waves of a predator-prey model with diffusion and nonlocal maturation delay

In this paper, a diffusive predator-prey system with nonlocal maturation delay is investigated. By analyzing the corresponding characteristic equations, the local stability of each of uniform steady states of the system is discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the system by using the method of upper–lower solutions and its associated monotone iteration scheme, respectively. The existence of travelling wave solution of the system is established by using the geometric singular perturbation theory. Numerical simulations are carried out to illustrate the theoretical results.     

Positive solutions for a modified Leslie–Gower prey–predator model with Crowley–Martin functional responses

In this paper, we study a modified Leslie–Gower prey–predator model with Crowley–Martin functional response. The stability and instability of the trivial and semi-trivial solutions was studied by analyzing the eigenvalues of the linearized system. The existence, multiplicity and uniqueness of positive steady state solutions were shown by using bifurcation theory, degree theory, energy estimate and asymptotic behavior analysis. Furthermore, all results were characterized in parameter plane.