Bifurcations in a white-blood-cell production model

We study the dynamics of a model of white-blood-cell (WBC) production. The model consists of two compartmental differential equations with two discrete delays. We show that from normal to pathological parameter values, the system undergoes supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles. We characterize the steady states of the system and perform a bifurcation analysis. Our results indicate that an increase in apoptosis rate of either hematopoietic stem cells or WBC precursors induces a Hopf bifurcation and an oscillatory regime takes place. These oscillations are seen in some hematological diseases.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

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.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Biomass and biodiversity in species-rich tritrophic communities

We consider a tritrophic system with one basal and one top species and a large number of primary consumers, and derive upper and lower bounds for the total biomass of the middle trophic level. These estimates do not depend on dynamical regime, holding for fixed point, periodic, or chaotic dynamics. We have two kinds of estimates, depending on whether the predator abundance is zero. All these results are uniform in a self-limitation parameter, which regulates prey diversity in the system. For strong self-limitation, diversity is large; for weak self-limitation, it is small. Diversity depends on the variance of species’ parameter values. The larger this variance, the lower the diversity, and vice versa. Moreover, variation in the parameters of the Holling type II functional response changes the bifurcation character, with the equilibrium state with nonzero predator abundance losing stability. If that variation is small then the bifurcation can lead to oscillations (the Hopf bifurcation). Under certain conditions, there exists a supercritical Hopf bifurcation. We then find a connection between diversity and Hopf bifurcations. We also show that the system exhibits top-down regulation and a hump-shaped diversity-productivity curve.We then extend the model by allowing species to experience self-regulation. For this extended model, explicit estimates of prey diversity are obtained. We study the dynamics of this system and find the following. First, diversity and system dynamics crucially depend on variation in species parameters. We show that under certain conditions, the system undergoes a supercritical Hopf bifurcation. We also establish a connection between diversity and Hopf bifurcations. For strong self-limitation, diversity is large and complex dynamics are absent. For weak self-limitation, diversity is small and the equilibrium with non-zero predator abundance is unstable.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

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.     

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.”     

Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model

The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.     

A delay reaction-diffusion model of the dynamics of botulinum in fish

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.     

Periodic oscillations in leukopoiesis models with two delays

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.     

Hopf bifurcation in three-species food chain models with group defense.

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.     

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.     

一类考虑收获的时滞捕食系统的稳定性与Hopf分支研究

给出了一类考虑收获的时滞捕食系统的局部稳定性判断,并由规范型理论和中心流形定理推导出了Hopf分支的方向、稳定性等条件,最后给出了两个数值模拟例子验证了结论的正确性.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

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.     

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     

一具有非线性发生率传染病模型的稳定性和霍普夫分支

在这篇文章中,我们研究了一具有非线性发生率的传染病模型．该模型经历了鞍结点分支和霍普夫分支．我们对模型的霍普夫分支进行了详细的分析,得知该霍普夫分支是超临界的．此外,我们给出了支持理论分析的数值模拟．     

Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks

We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model.