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.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Stability and bifurcation analysis of a nutrient-phytoplankton model with time delay

We proposed a nutrient-phytoplankton interaction model with a discrete and distributed time delay to provide a better understanding of phytoplankton growth dynamics and nutrient-phytoplankton oscillations induced by delay. Standard linear analysis indicated that delay can induce instability of a positive equilibrium via Hopf bifurcation. We derived the conditions guaranteeing the existence of Hopf bifurcation and tracked its direction and the stability of the bifurcating periodic solutions. We also obtained the sufficient conditions for the global asymptotic stability of the unique positive steady state. Numerical analysis in the fully nonlinear regime showed that the stability of the positive equilibrium is sensitive to changes in delay values under select conditions. Numerical results were consistent with results predicted by linear analysis.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate 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.     

Global stability and Hopf bifurcation of a plankton model with time delay

A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter.     

Effect of time delay in an autotroph-herbivore system with nutrient cycling

In the present study we consider a mathematical model of a non-interactive type autotroph-herbivore system in which the amount of autotroph biomass consumed by the herbivore is assumed to follow a Holling type II functional response. We have also incorporated discrete time delays in the numerical response term to represent a delay due to gestation, and in the recycling term which represents the time required for bacterial decomposition. We have derived conditions for global asymptotic stability of the model in the absence of delays. Conditions for delay-induced asymptotic stability of the steady state are also derived. The length of the delay preserving stability has been estimated and interpreted ecologically.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.     

Stability and bifurcations for the chronic state in Marchuk's model of an immune system

In the paper we present known and new results concerning stability and the Hopf bifurcation for the positive steady state describing a chronic disease in Marchuk's model of an immune system. We describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system. We compare these results with the results known in the literature. We show that the positive steady state can be stable only for very specific parameter values. Stability analysis is illustrated by Mikhailov's hodographs and numerical simulations. Conditions for the Hopf bifurcation and stability of arising periodic orbit are also studied. These conditions are checked for arbitrary chosen realistic parameter values. Numerical examples of arising due to the Hopf bifurcation periodic solutions are presented.     

Global Qualitative Analysis for a Ratio-Dependent Predator-Prey Model with Delay

A ratio-dependent predator-prey model with time lag for predator is proposed and analyzed. Mathematical analyses of the model equations with regard to boundedness of solutions, nature of equilibria, permanence, and stability are analyzed. We note that for a ratio-dependent system local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system and, therefore, does not imply global asymptotic stability. It is found that an orbitally asymptotically stable periodic orbit exists in that model. Some sufficient conditions which guarantee the global stability of positive equilibrium are given.     

基于比率的广义Holling-Tanner系统的全局渐近稳定性

本文对基于比率的广义Holling-Tanner系统进行了定性分析,给出了系统解有界的条件和平衡点局部稳定的条件．通过构造适当的Liapunov函数,得到了正平衡点全局渐近稳定性的充分条件．     

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.     

Monotone methods and stability results for nonlocal reaction-diffusion equations with time delay

In this paper, we study the applications of the monotone iteration method for investigating the existence and stability of solutions to nonlocal reaction-diffusion equations with time delay. We emphasize the importance of the idea of monotone iteration schemes for investigating the stability of solutions to such equations. We show that every steady state of such equations obtained by using the monotone iteration method is priori stable and all stable steady states can be obtained by using such method. Finally, we apply our main results to three population models.