On the existence of limit cycles in motion field

In this paper we show the existence of limit cycles in the vector field generated by the perspective projection on the image plane of the velocity field of a moving surface. The existence of limit cycles is proved with the Poincaré-Bendixon theorem, in the case of a rotating smooth non-planar surface and illustrated with computer graphics. The structural stability of the limit cycles is also discussed.     

Qualitative analysis of a predator-prey system with limit cycles

Fairly regular multiannual microtine rodent cycles are observed in boreal Fennoscandia. In the southern parts of Fennoscandia these multiannual cycles are not observed. It has been proposed that these cycles may be stabilized by generalist predation in the south.We show that if the half-saturation of the generalist predators is high compared to the number of small rodents the cycles are likely to be stabilized by generalist predation as observed. We give examples showing that if the half-saturation of the generalist predators is low compared to the number of small rodents, then multiple equilibria and multiple limit cycles may occur as the generalist predator density increases.     

Multiple limit cycles in the standard model of three species competition for three essential resources

We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside' one is an unstable separatrix and the ``outside' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix. This research was partially supported by NSF grant DMS 0211614. KY 40292, USA. This author's research was supported in part by NSF grant DMS 0107160     

On the stability of metabolic cycles

We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.     

On the analysis of substrate cycles in large metabolic systems

The simultaneous operation of paired, opposing reactions (substrate cycles) or parallel reactions (dual pathways) with seeming wastage of ATP is widespread in cellular metabolism. Analysis of such “futile” pathways has hitherto been limited to loci with only two or three interconnecting fluxes. We introduce here a method that allows straightforward analysis of more complex systems. The method involves the linear superposition of “fundamental” modes, one or more of which may be energetically wasteful. Decomposition of a flux pattern into such modes allows computation of the amount of free energy “wasted” at any locus. Appropriate normalizations of energy wastage yield a number of indices useful for assessing the energetic impact of futile pathways on the cell and for comparing the degree of regulation of substrate cycles or dual pathways under different metabolic conditions. This approach is applied to steady-state flux data obtained in the protozoanTetrahymena pyriformis and in isolated rat hepatocytes under a variety of conditions.     

Spatially extended host-parasite interactions: the role of recovery and immunity

Techniques for determining the long-term dynamics of host-parasite systems are well established for mixed populations. The field of spatial modelling in ecology is more recent but a number of key advances have been made. In this paper, we use state-of-the-art approximation techniques, supported by simulations, in order to investigate the role of recovery and immunity in spatially structured populations. Our approach is to use correlation models, namely pair-wise models, to capture the spatial relationships of contacts and interactions between individuals. We use the pair-wise framework to address a number of key ecological questions; including, the persistence of endemic limit cycles and regions of parasite-driven extinction--features which differentiate spatial from non-spatial models--and the effects on invasion fitness. We demonstrate a loss of limit cycle behaviour, in addition to an increase in the critical transmissibility and extinction thresholds, when recovery is included. This approach allows for a better analytical understanding of the dynamics of host-parasite interactions and demonstrates the importance of recovery and immunity in local interactions.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

生化反应中一类三次系统的极限环

研究生化反应中一类三次系统：dx/dt=-x-Ф1（x）＋yФ2（x）,dy/dt=a0＋Ф1（x）-yФ2（x）其中Ф1（x）=Ax^3＋ax^2＋bx＋B,Ф2（x）=cx^2＋dx＋e．较完整地解决了该系统极限环的存在性,唯一性与不存在性等问题．     

Doublet frequencies in the DNA of genetic code limit organisms

Summary Unrelated organisms with DNA of extreme G + C content (25% or 70%) are found to share very specific patterns of nearest neighbour base doublet frequency in their DNAs. This is shown to be a result of restrictions on the extremity of amino acid composition in their proteins, combined with a maximisation of the use of one type of base pair in redundant codon positions. Inferences are made about the universal nature of the genetic code and the proportion of DNA used for specifying protein in different species. The composition of coding DNA strands in these organisms is also discussed.     

On the optimum distribution of enzyme feed in a cascade of CSTR's performing an enzyme-catalyzed reaction with deactivation

A search for the optimum fractional distribution of an enzyme-rich stream to the various reactors of a cascade of CSTR's was implemented. A theoretical analysis, laid out in dimensionless form and based on the assumptions that the system is operated under steady state conditions, the enzyme undergoes first order deactivation, and the reaction catalyzed by the enzyme follows Michaelis-Menten kinetics, is reported. The objective function utilised is the minimisation of the overall volume of the cascade, and analytical expressions are obtained for the concentration of active enzyme and substrate in the outlet stream from each reactor. It is found that the best option is to add the whole enzyme-rich stream to the first reactor in the cascade irrespective of the operating and kinetic parameters of the system.     

Neural representation of the acoustic biotope: On the existence of stimulus-event relations for sensory neurons

In the context of a behaviourally oriented investigation of neural information processing we have studied single unit responses from cochlear nucleus (12 units) and auditory cortex (23 units) of the cat. Attention was focused on two aspects: the composition of a wide stimulus ensemble and the influence of this ensemble on the single unit neural activity: action potentials or neural events. The stimulus ensemble was based on sounds occurring in the natural environment of the animal: the Acoustic Biotope. A realization of such an ensemble was constructed and used in the experiments. A systematic analysis of the effect of this stimulus ensemble on the neural activity comprises different questions with respect to a possible stimulus-event relation: existence and nature. In the present paper emphasis is given to the question of existence of a s-e relation. The existence of such a relation has been investigated by presenting the stimulus ensemble twice. The degree of reproducibility of neural activity indicates to what extent a s-e relation indeed exists. A formal measure of testing the reproducibility of two sequences of neural events is defined: the Cross Coincidence Function. Results are presented, interpretation and formal problems discussed. Results obtained so far indicate that on the whole the existence of a s-e relation can be established more definitely for cochlear nucleus units than for auditory cortex units. Applications of the presented procedure arise in exploratory investigations in central regions of the sensory nervous system and in regions with presumed multimodal inputs.     

两种群均具有常收获率或常投放率的Lotka-Volterra系统可以存在三个极限环

证明了两种群均具有常收获率或常投放率的Lotka-Volterra系统可以存在三个极限环。     

Further evidence for the existence of a membrane potential in Trypanosoma brucei brucei

The distribution of 137Cs+, in the presence of valinomycin, has been used to measure the magnitude of the membrane potential (delta psi) in bloodstream forms of Trypanosoma brucei brucei. Values of the delta psi falling in the range -100 mV to -160 mV were observed and the maintenance of this delta psi was sensitive to certain ionophores and protonophores.     

Female reproductive cycles in two subspecies of the tropical lizard Mabuya striata

Summary Female reproductive cycles were examined in two subspecies of the live-bearing lizard Mabuya striata from Central Africa, an area with distinct rainy and dry seasons. The low altitude M. s. striata was reproductive throughout the year apart from a brief period at the start of the rainy season, and probably produced three clutches a year. Most females of the high altitude M. s. punctatissima were non-reproductive in the rainy season, came into reproductive condition in the early dry season, and gave birth in the late dry season. For some females, there was a second reproductive cycle starting in the late dry season with birth in the rainy season. For the low altitude M. s. striata it appears that availability of moisture may affect reproduction. For the high altitude M. s. punctatissima it appears that the low temperatures and short hours of sunshine of the early dry season constrain reproduction, and that reproductive cycles are timed to avoid birth occurring in the early dry season.     

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     

冻融环境下凋落叶添加对亚高山森林土壤腐殖化程度的影响

亚高山森林凋落叶与土壤腐殖化的关系是了解森林土壤有机质动态过程的关键内容,且受到季节性冻融、凋落叶与土壤性质的综合作用,但缺乏深入认识。因此,在对亚高山凋落叶腐殖化认识的前期基础上,以川西亚高山典型的针叶林、针阔混交林以及阔叶林土壤为研究对象,采用控制冻融环境和凋落叶添加的方法,通过分析土壤可提取腐殖质的色调系数(Δlog K)、光密度值(E4/E6)以及A600/C值等光学指标,研究冻融循环与凋落叶添加对土壤腐殖化程度的影响。结果表明:冻融循环和凋落叶添加及其交互作用均明显影响土壤的腐殖化程度,但受到凋落叶类型和土壤类型的调控。在冻融循环初期,添加凋落叶能促进土壤的腐殖化进程,提高土壤的腐殖化程度;随着冻融循环次数的增加以及凋落叶的不断分解,阔叶凋落叶对土壤的腐殖化表现出促进作用,而针叶凋落叶则对土壤的腐殖化表现出抑制作用。此外,在冻融循环环境下,即使土壤的腐殖化程度增加,但腐殖质仍为相对年轻的Rp型,有利于提高土壤的肥力。这些结果表明,气候变化情景下高山森林地表冻融循环的改变将显著影响凋落叶与土壤腐殖质的密切关系,为进一步认识区域森林土壤有机质动态提供了一定的科学依据。