Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.     

Foraging facilitation among predators and its impact on the stability of predator–prey dynamics

Predator foraging facilitation may strongly influence the dynamics of a predator–prey system. This behavioral pattern is well-observed in real life interactions, but less is known about its possible impacts on the predator–prey dynamics. In this paper we analyze a modified Rosenzweig–MacArthur model, where a predator-dependent family of functions describing predator foraging facilitation is introduced into the Holling type II functional response. As the general assumption of foraging facilitation is that higher predator densities give rise to an increased foraging efficiency, we model predator facilitation with an increasing encounter rate function. Using the tools of bifurcation analysis we describe all the nonlinear phenomena that occur in the system provoked by foraging facilitation, these include the fold, Hopf, transcritial, homoclinic and Bogdanov–Takens bifurcation. We show that foraging facilitation can stabilize the coexistence in the predator–prey system for specific rates, but in most of the cases it can have fatal consequences for the predators themselves.     

Arterial bifurcations in the human retina

The branching angles and relative diameters of blood vessels in 51 arterial bifurcations in the retina of a normal human eye were measured. In eight other bifurcations, only the total branching angles were measured. The results are compared with theoretical predictions in an attempt to understand the physiological principles governing branching in the cardiovascular system.     

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.     

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.     

The effect of predator limitation on the dynamics of simple food chains

We investigate the influence of competition between predators on the dynamics of bitrophic predator–prey systems and of tritrophic food chains. Competition between predators is implemented either as interference competition, or as a density-dependent mortality rate. With interference competition, the paradox of enrichment is reduced or completely suppressed, but otherwise, the dynamical behavior of the systems is not fundamentally different from that of the Rosenzweig–MacArthur model, which contains no predator competition and shows only continuous transitions between fixed points or periodic oscillations. In contrast, with density-dependent predator mortality, the system shows a surprisingly rich dynamical behavior. In particular, decreasing the density regulation of the predator can induce catastrophic shifts from a stable fixed point to a large oscillation where the predator chases the prey through a cycle that brings both species close to the threshold of extinction. Other catastrophic bifurcations, such as subcritical Hopf bifurcations and saddle-node bifurcations of limit cycles, do also occur. In tritrophic food chains, we find again that fixed points in the model with predator interference become unstable only through Hopf bifurcations, which can also be subcritical, in contrast to the bitrophic situation. The model with a density limitation shows again catastrophic destabilization of fixed points and various nonlocal bifurcations. In addition, chaos occurs for both models in appropriate parameter ranges.     

一类具有时滞的传染病模型的稳定性分析

研究了一类具有时滞的传染病生物模型．首先研究了该模型的线性稳定性,并给出了一列Hopf分支值,然后利用中心流形定理和正规型方法,给出了确定分支周期解的分支方向与稳定性的计算公式．     

Nonsymmetrical bifurcations in arterial branching

The results of optimality studies of the branching angles of arterial bifurcations are extended to nonsymmetrical bifurcations. Predicted nonsymmetrical bifurcations are found to be not unlike those observed in the cardiovascular system.     

Transient spatio-temporal dynamics of a diffusive plant–herbivore system with Neumann boundary conditions

In many existing predator–prey or plant–herbivore models, the numerical response is assumed to be proportional to the functional response. In this paper, without such an assumption, we consider a diffusive plant–herbivore system with Neumann boundary conditions. Besides stability of spatially homogeneous steady states, we also derive conditions for the occurrence of Hopf bifurcation and steady-state bifurcation and provide geometrical methods to locate the bifurcation values. We numerically explore the complex transient spatio-temporal behaviours induced by these bifurcations. A large variety of different types of transient behaviours including oscillations in one or both of space and time are observed.     

Classification of dynamics of a model of motor coordination and comparison with Parkinson's disease data

In our recent reports motor coordination of human lower limbs has been investigated during pedaling a special kind of ergometer which allows its left and right pedals to rotate independently. In particular, relative phase between left and right rotational-velocity waveforms of the pedals and their amplitude modulation have been analyzed for patients with Parkinson's disease (PD). Several patients showed peculiar interlimb coordination different from the regular anti-phase pattern of normal subjects. We have reported that these disordered patterns could be classified into four groups. Moreover, it has been demonstrated that a mathematical model could reproduce most of the disordered patterns. Such a model includes a schematization of the central pattern generator with two identical half-centers mutually coupled and two tonic control signals from higher motor centers, each of which inputs to one of the half-centers. Depending on the intensities of the tonic signals and on the differences between them, the model could generate a range of dynamics comparable to the clinically observed disordered patterns. In this paper, we explore the dynamics of the model by varying the intensities of the tonic signals in the model. Using the same method used for classifying the clinical data, the dynamics of the model are classified into several groups. The classified groups for the simulated data are compared with those for the clinical data to look at qualitative correspondence. Our systematic exploration of the model's dynamics in a wide range of the parameter space has revealed global organization of the bifurcations including Hopf bifurcations and cascades of period-doubling bifurcations among others, suggesting that the bifurcations, induced by instability of stable dynamics of the human motor control system, are responsible for the emergence of the disordered coordination in PD patients.     

Bifurcations of large networks of two-dimensional integrate and fire neurons

Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by Touboul (SIAM J Appl Math 68(4):1045–1079, 2008). However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.     

The dynamics of two-species allelopathic competition with optimal harvesting

This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare–Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

具有三时滞Lotka-Volterra互惠系统的Hopf分支

建立了具有三个时滞的Lotka-Volterra互惠系统;获得了正平衡点和Hopf分支存在的条件等;并对所获得的结果进行了数值模拟．     

Arterial branching in man and monkey

Vessel diameters and branching angles are measured from a large number of arterial bifurcations in the retina of a normal human subject and in that of a rhesus monkey. The results are compared with each other and with theoretical results on this subject.     

A predator-prey reaction-diffusion system with nonlocal effects

We consider a predator-prey system in the form of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.     

Local geometry of arterial branching

A model of the geometrical structure of arterial bifurcations is proposed in the context of optimality of the bifurcation as a fluid conducting system. Optimality is considered both globally, in terms of the cardiovascular system as a whole, and locally, in terms of the orderliness of the flow in the bifurcation region. It is shown that a bifurcation can be optimal both globally and locally. Typical examples of such bifurcations are given.     

Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system

The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus. The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards. The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution.     

Dynamics and bifurcations of a discrete-time prey-predator model with Allee effect on the prey population

This paper studies the dynamic behavior of a discrete-time prey-predator model. It is shown that this model undergoes codimension one and codimension two bifurcations such as transcritical, flip (period-doubling), Neimark-Sacker and strong resonances 1:2, 1:3 and 1:4. The bifurcation analysis is based on the numerical normal form method and the bifurcation scenario around the bifurcation point is determined by their critical normal form coefficients. The advantage of this method is that there is no need to calculate the center manifold and to convert the linear part of the map to a Jordan form. The bifurcation curves of fixed points under variation of one and two parameters are obtained, and the codimensions one and the two bifurcations on the corresponding curves are computed.     

On bifurcations in complex ecological systems with diffusion and noise

We investigate here several methods for the qualitative investigation of complex ecological systems including diffusion and noise. Assuming that the systems are described by diffusion equations or Fokker–Planck equations, we formulate for gradient-type dynamical systems several statements about the number and type of possible bifurcations using theorems of catastrophe theory. We introduce stochastic potentials, present exact and approximate solutions. In the case of general dynamical systems we show that the case of strong noise may be appropriately described by expansions of the stochastic potential with respect to reciprocal noise. Finally we present examples for the bifurcations of the biomass.