On local bifurcations in neural field models with transmission delays

Neural field models with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.     

Stability of periodic solutions for a model of genetic repression with delays

A technique is discussed for locating the Hopf bifurcation of an n-dimensional system of delay differential equations which arises from a model for control of protein biosynthesis. Certain parameter values are shown to allow a Hopf bifurcation to periodic orbits. At the Hopf bifurcation the periodic orbits are shown to be stable either analytically or numerically depending on the parameter values.On leave from North Carolina State University.Supported in part by N.S.F. Grant # MCS 81-02828     

Dynamical analysis of a diffusion plant-wrack model with delay

In this paper, in view of the senescence of plant and the decay of wrack, time delays are introduced into the plant-wrack model. The effects of wrack decay and time delay on the dynamical behaviors of the diffusive plant-wrack model are studied analytically and numerically. When the delay is zero, the wrack decay will induce the change of stability of the unique equilibrium point, further lead to the occurrence of the Hopf bifurcation and the Turing instability. When the delay is present, the conditions for the occurrence of the Hopf bifurcation are established. By comparing the results of the model without and with delay, it is found that the increases of delay may induce no stability switches, a single stability switch or multiple stability switches, when the value of wrack decay can stabilize model with zero delay. When the value of wrack decay can destabilize model with zero delay, numerical simulations show that the small delay may cause homogeneous distributions of vegetation, while the larger delay may cause the emergence of periodic oscillation of vegetation. The obtained results provide a basis for understanding the spatiotemporal evolution of such a plant-wrack model with delay.     

Oscillatory viral dynamics in a delayed HIV pathogenesis model

We consider an HIV pathogenesis model incorporating antiretroviral therapy and HIV replication time. We investigate the existence and stability of equilibria, as well as Hopf bifurcations to sustained oscillations when drug efficacy is less than 100%. We derive sufficient conditions for the global asymptotic stability of the uninfected steady state. We show that time delay has no effect on the local asymptotic stability of the uninfected steady state, but can destabilize the infected steady state, leading to a Hopf bifurcation to periodic solutions in the realistic parameter ranges.     

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction–diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction–diffusion system is Baer and Rinzel’s continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon “complete accommodation.”     

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

Periodicity in an epidemic model with a generalized non-linear incidence

We develop and analyze a simple SIV epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized non-linear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the Poincaré index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate. Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.     

一类时滞神经网络模型的稳定性与全局Hopf分支

研究了一类由两个神经元构成的时滞神经网络模型的稳定性和局部Hopf分支,并结合一般泛函微分方程的全局Hopf分支定理,利用度理论研究了全局Hopf分支的存在性．     

Stability switches,oscillatory multistability,and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.     

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

一类具有暂时免疫传染病模型的Hopf分支

本文应用Hassard的“规范形”方法,讨论了一类具有暂时免疫传染病模型的动力形态．给出了该系统发生Hopf分支的参数曲线,进一步计算出了决定分支方向及稳定性的参数条件,并给出了生态解释．     

Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.     

Stage-structured cannibalism with delay in maturation and harvesting of an adult predator

A three-dimensional stage-structured predator–prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.     

Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.     

Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington–DeAngelis incidence and three delays

In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington–DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.     

Delay-induced model for tumor-immune interaction and control of malignant tumor growth

The paper deals with the qualitative analysis of the solutions of a system of delay differential equations describing the interaction between tumor and immune cells. The asymptotic stability of the possible steady states is showed and the occurrence of limit cycle of the system around the interior equilibrium is proved by the application of Hopf bifurcation theorem by using the delay as a bifurcation parameter. The length of the delay parameter for preserving stability of the system is also estimated, which gives the idea about the mode of action for controlling oscillations in malignant tumor cell growth. The theoretical and numerical outcomes have been supported through experimental results from literatures. This approach gives new insight of modeling tumor-immune interactions and provides significant control strategies to overcome the large oscillations in tumor cells.     

Symmetry Breaking in a Model of Antigenic Variation with Immune Delay

Effects of immune delay on symmetric dynamics are investigated within a model of antigenic variation in malaria. Using isotypic decomposition of the phase space, stability problem is reduced to the analysis of a cubic transcendental equation for the eigenvalues. This allows one to identify periodic solutions with different symmetries arising at a Hopf bifurcation. In the case of small immune delay, the boundary of the Hopf bifurcation is found in a closed form in terms of system parameters. For arbitrary values of the time delay, general expressions for the critical time delay are found, which indicate bifurcation to an odd or even periodic solution. Numerical simulations of the full system are performed to illustrate different types of dynamical behaviour. The results of this analysis are quite generic and can be used to study within-host dynamics of many infectious diseases.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel