一类两种群均有收获率捕食系统的细焦点与极限环

研究一类具有HollingⅡ类功能反应且两种群均为非常数收获率的捕食系统,其中食饵种群具有非线性密度制约.利用微分方程定性与稳定性理论及分支理论,得到系统平衡点的性态及极限环存在与否的充分条件,利用Hopf分支理论得到存到多个极限环的充分条件.     

Analytic procedures for large dimention nonlinear biochemical oscillators.

An analytic method is presented which can be used to determine if the following system of nonlinear differential equations has periodic solutions x1 = h(xn)-b1x1 xj = gj-1xj-1-bjxj j = 2, ... n A systematic dual input describing function procedure is given for constructing a function of the reaction constants R, where if R greater than 1 a periodic solution exists and if R smaller than 1 there is no periodic solution. The form of R constructed generalizes immediately to an arbitrarily large dimension. The method generalizes to cover systems displaying hysteresis kinetics, systems subject to chemical noise, and systems containing delay components. The method has been applied to a well known biochemical problem where h(xn)-k/(1 + alphaxnrho). For rho = 1, for all n, there are no stable limit cycles such that xj(t) greater than O, t larger than or equal to O. For rho = 2,n larger than or equal to 8 it is possible to construct a parameter set such that stable oscillations appear.     

An analysis of models describing predator-prey interaction

Mathematical models of the interaction between predator and host populations have been expressed as systems of nonlinear ordinary differential equations. Solutions of such systems may be periodic or aperiodic. Periodic, oscillatory solutions may depend on the initial conditions of the system or may be limit cycles. Aperiodic solutions can, but do not necessarily, exhibit oscillatory behavior. Therefore, it is important to characterize predatory-prey models on the basis of the possible types of solutions they may possess. This characterization can be accomplished using some well-known methods of nonlinear analysis. Examination of the system singular points and inspection of phase plane portraits have proved to be useful techniques for evaluating the effect of various modifications of early predator-prey models. Of particular interest is the existence of limit cycle oscillations in a model in which predator growth rate is a function of the concentration of prey.     

可逆两分子饱和反应动力系统的极限环

研究生化反应中一类可逆两分子饱和反应的数学模型dx/dt＝δ-xy cy^2,dy/dt=xy-cy^2-ay/b b,应用微分方程定性理论,完整地解决了该系统极限环的存在性,不存在性和唯一性。     

Uniqueness of limit cycles in models for microparasitic and macroparasitic diseases

 The aim of this paper is to prove the uniqueness of isolated periodic solutions (i.e. limit cycles) in two simple models for microparasitic and macroparasitic diseases. Both models are described by systems of planar autonomous ordinary differential equations. After transformation of these systems to generalized Liénard systems, we will apply a modified theorem of Zhang and Dulac’s criterion to prove the uniqueness of limit cycles. Received 27 February; received in revised form 19 May 1997     

A competition model with dynamically allocated inhibitor production

The chemostat is a basic model for competition in an open system and a model for the laboratory bio-reactor (CSTR). Inhibitors in open systems are studied with a view of detoxification in natural systems and of control in bio-reactors. This study allows the amount of resource devoted to inhibitor production to depend on the state of the system. The feasibility of one dependence is provided by quorum sensing. In contrast to the constant allocation case, a much wider set of outcomes is possible including interior, stable rest points and stable limit cycles. These outcomes are important contrasts to competitive exclusion or bistable attractors that are often the outcomes for competitive systems. The model consists of four non-linear ordinary differential equations and computer software is used for most of the stability calculations.     

Algorithmic Global Criteria for Excluding Oscillations

We investigate algorithmic methods to tackle the following problem: Given a system of parametric ordinary differential equations built by a biological model, does there exist ranges of values for the model parameters and variables which are both meaningful from a biological point of view and where oscillating trajectories, can be found? We show that in the common case of polynomial vector fields known criteria excluding the existence of non-constant limit cycles lead to quantifier elimination problems over the reals.We apply these criteria to various models that have been previously investigated in the context of algebraic biology.     

一类多分子生化反应系统的极限环

应用微分方程定性理论,研究了生化反应中一类多分子一级饱和反应的数学模型的极限环的存在性、不存在性和唯一性问题．     

一类可逆多分子饱和生化反应系统的非线性分析

研究生化反应中一类可逆多分子饱和反应系统x=a-xy^n cy^n 1,y=xy^n-cy^n 1-dy／(y 6),应用微分方程定性理论,完整的解决了该系统极限环的存在性、不存在性和唯一性问题。     

Coexistence of competing microbial species in a chemostat where one population feeds on another

In this paper, we consider a model for a chemostat in which two microbial species compete for a single rate-limiting nutrient, while one of the species feeds on another. Under certain simplifying hypotheses, such a chemostat can be described by a system of three nonlinear ordinary differential equations. A theoretical study is conducted to characterize the possible types of solutions. A limit cycle solution was obtained for some parametric values of the system indicating that coexistence of the two species is possible in a significant range of the operating parameters.     

一个稀疏效应下的Volterra系统的极限环

应用数学生态学和微分方程定性理论,讨论了一个稀疏效应下的Volterra系统,在给定参数满足一定的条件下,证明了该系统极限环的存在性和唯一性,以及该系统的正平衡点全局渐近稳定。     

An efficient simulation method for discrete-value controlled large-scale neuromyoskeletal system models

An efficient Euler-Adams hybrid integration scheme for simulating on the computer discrete-value controlled large-scale neuromyoskeletal system models is presented. If, as discussed in the model, the differential equations describing the recruitment and excitation dynamics of the muscular subsystem are independent of the corresponding contraction-dynamical state variables, they can be integrated separately over certain time intervals by a modified Euler routine that handles discontinuous right-hand sides efficiently. The resulting myostates can then be stored and used as continuous input values for the subsequent integration by an Adams predictor-corrector algorithm of the remaining contraction-dynamical and skeletomechanical state differential equations. With such an Euler-Adams hybrid integration routine one avoids the detrimental effects and efficiency losses associated with frequent stop-restart cycles of otherwise efficient Adams-type algorithms, which cycles are forced by discontinuities on the right-hand side of the myostate equations. In the example presented, a reduction in the execution time by a factor of about 5 could be achieved by implementing the proposed technique.     

Small autocatalytic reaction networks—III. Monotone growth functions

The classification of the dynamical behaviour of first order replicator equations is extended to models with monotonical growth rates. It is shown that for two species there is a general classification independent of the particular form of the growth function. For three species a common dynamical behaviour for all power laws can be found and the existence of limit cycles is disproved. For more general growth functions, however, limit cycles may occur.     

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.     

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.     

Solutions to axon equations

The solutions to a general class of axon partial differential equations proposed by FitzHugh which includes the Hodgkin-Huxley equations are studied. It is shown that solutions to the partial differential equations are exactly the solutions to a related set of integral equations. An iterative procedure for constructing the solutions based on standard methods for ordinary differential equations is given and each set of initial values is shown to lead to a unique solution. Continuous dependence of the solutions on the initial values is established and solutions with initial values in a restricted (physiological) range are shown to remain in that range for all time. The iterative procedure is not suggested as the basis for numerical integration.     

Limit cycles of the two-dimensional motion field

Recently, it has been shown that there can be limit cycles in the vector field generated by the perspective projection on the image plane of the three dimensional velocity field of a certain class of non-planar rotating surfaces. In this paper, it is shown that, for any possible rigid motion, there cannot be limit cycles in the motion field of a planar surface. Therefore, the presence of limit cycles in the motion field is necessarily due to the non-planar structure of the viewed scene. An experiment on real images is also presented in which a limit cycle occurs when two planar patches have different orientation in space rotate around a fixed axis.     

Stable periodic solutions for two species,density dependent coevolution

This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis-Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.     

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.     

The effects of time delays in a phosphorylation-dephosphorylation pathway

Complex signaling cascades involve many interlocked positive and negative feedback loops which have inherent delays. Modeling these complex cascades often requires a large number of variables and parameters. Delay differential equation models have been helpful in describing inherent time lags and also in reducing the number of governing equations. However the consequences of model reduction via delay differential equations have not been fully explored. In this paper we systematically examine the effect of delays in a complex network of phosphorylation-dephosphorylation cycles (described by Gonze and Goldbeter, J. Theor. Biol., 210, (2001) 167-186), which commonly occur in many biochemical pathways. By introducing delays in the positive and negative regulatory interactions, we show that a delay differential model can indeed reduce the number of cycles actually required to describe the phosphorylation-dephosphorylation pathway. In addition, we find some of the unique properties of the network and a quantitative measure of the minimum number of delay variables required to model the network. These results can be extended for modeling complex signalling cascades.