一类分数阶微分方程的最优控制

主要研究了一类分数阶微分方程的最优控制问题.通过Oustaloup迭代逼近,可以将分数阶微分算子在频率域范围内进行近似.那么原先的分数阶微分方程就转换为一般的常微分方程.利用这个关系,可以得到关于分数阶微分方程的两个定理和一个引理.最后给出一个例子说明该方法的有效性.     

分数阶微分方程边值问题正解的唯一性

分数阶微分方程在生物科学和其他领域发挥着重要作用.考虑来类分数阶微分方程边值问题正解的这来性,得到的结证不仅可以保证正解的存在这来,而且可以用于构造迭代序列来逼近这个解.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

线性双滞量NDDE振动性的代数判据

本文给出一阶双滞量NDDE对所有参数振动与非振动的充要条件，这些条件用系统诸参数的若干代数式表达，方便于应用验证．     

Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: how often should you vaccinate and how strong should the vaccine be?

To stimulate the immune system's natural defenses, a post-infection HIV vaccination program to regularly boost cytotoxic T-lymphocytes has been proposed. We develop a mathematical model to describe such a vaccination program, where the strength of the vaccine and the vaccination intervals are constant. We apply the theory of impulsive differential equations to show that the model has an orbitally asymptotically stable periodic orbit, with the property of asymptotic phase. We show that, on this orbit, the vaccination frequency can be chosen so that the average number of infected CD4(+) T cells can be made arbitrarily low. We illustrate the results with numerical simulations and show that the model is robust with respect to both the parameter choices and the formulation of the model as a system of impulsive differential equations.     

具体长结构的鱼类种群的数学模型及其应用

本文报道了一个新的鱼类种群的数学模型,它是一阶变系数线性偏微分方程。该模型可用于预测某一水域中某种鱼类种群不同体长的鱼的数量随时间变化的情况。       

卡尔文循环的数学模型

本文运用常微分方程组的数学方法,建立了卡尔文循环的数学模型。     

Experimental manipulation of female tail length did not cause differential allocation by males in the barn swallow

The evolution and maintenance of female ornamentation has attracted increasing attention, because the previous explanation, that is a non‐functional copy of functional male ornamentation, seems insufficient to explain female ornamentation. A post‐mating sexual selection, differential allocation, may be more common than pre‐mating sexual selection, but few studies have investigated differential allocation by males. Here, we studied differential allocation of incubation investment by male barn swallows Hirundo rustica, a model species for the study of sexual selection, because our previous correlative study demonstrated a positive relationship between female tail length and male incubation investment. We manipulated the length of the outermost tail feathers in females after clutch completion and examined whether males adjust incubation investment according to female ornamentation. Because extra‐pair paternity is virtually absent in the study population, we were able to study differential allocation based on the tradeoff between current and future reproductive investments, rather than the tradeoff between current paternal investment and additional mating effort. The experimental treatment had no significant effect on male nest attentiveness, whereas female tail length before manipulation predicted male nest attentiveness. The observed pattern is consistent with differential access; that is, well‐ornamented individuals have greater access to mates with high reproductive (parental) ability, rather than differential allocation during incubation. Alternatively, males can directly assess eggs in their nests, and thus, as seen in other species, males might adjust their incubation investment based on the egg characteristics of long‐tailed females.     

An implicit approach to model plant infestation by insect pests

Various spatial approaches were developed to study the effect of spatial heterogeneities on population dynamics. We present in this paper a flux-based model to describe an aphid-parasitoid system in a closed and spatially structured environment, i.e. a greenhouse. Derived from previous work and adapted to host-parasitoid interactions, our model represents the level of plant infestation as a continuous variable corresponding to the number of plants bearing a given density of pests at a given time. The variation of this variable is described by a partial differential equation. It is coupled to an ordinary differential equation and a delay-differential equation that describe the parasitized host population and the parasitoid population, respectively. We have applied our approach to the pest Aphis gossypii and to one of its parasitoids, Lysiphlebus testaceipes, in a melon greenhouse. Numerical simulations showed that, regardless of the number and distribution of hosts in the greenhouse, the aphid population is slightly larger if parasitoids display a type III rather than a type II functional response. However, the population dynamics depend on the initial distribution of hosts and the initial density of parasitoids released, which is interesting for biological control strategies. Sensitivity analysis showed that the delay in the parasitoid equation and the growth rate of the pest population are crucial parameters for predicting the dynamics. We demonstrate here that such a flux-based approach generates relevant predictions with a more synthetic formalism than a common plant-by-plant model. We also explain how this approach can be better adapted to test different management strategies and to manage crops of several greenhouses.     

Modelling animal growth in random environments: An application using nonparametric estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data.     

A Simple Unforced Oscillatory Growth Model in the Chemostat

In a chemostat, transient oscillations in cell number density are often experimentally observed during cell growth. The aim of this paper is to propose a simple autonomous model which is able to generate these oscillations, and to investigate it analytically. Our point of view is based on a simplification of the cell cycle in which there are two states (mature and immature) with the transfer between the two dependent on the available resources. We use the mathematical global properties of competitive differential systems to prove the existence of a limit cycle. A comparison between our model and a more complex model consisting of partial differential equations is made with the help of numerical simulations, giving qualitatively similar results.     

Applying Methods from Differential Geometry to Devise Stable and Persistent Air Layers Attached to Objects Immersed in Water

We describe a few mathematical tools which allow to investigate whether air-water interfaces exist(under prescribedconditions)and are mechanically stable and temporally persistent.In terms of physics,air-water interfaces are governed by theYoung-Laplace equation.Mathematically they are surfaces of constant mean curvature which represent solutions of a nonlinearelliptic partial differential equation.Although explicit solutions of this equation can be obtained only in very special cases,it is-under moderately special circumstances-possible to establish the existence of a solution without actually solving thedifferential equation.We also derive criteria for mechanical stability and temporal persistence of an air layer.Furthermorewe calculate the lifetime of a non-persistent air layer.Finally,we apply these tools to two examples which exhibit the symmetriesof 2D lattices.These examples can be viewed as abstractions of the biological model represented by the aquatic fernSalvinia.     

Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks

Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models. We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model     

Mathematical model for adaptive evolution of populations based on a complex domain

A mutation is ultimately essential for adaptive evolution in all populations. It arises all the time, but is mostly fixed by enzymes. Further, most do consider that the evolution mechanism is by a natural assortment of variations in organisms in line for random variations in their DNA, and the suggestions for this are overwhelming. The altering of the construction of a gene, causing a different form that may be communicated to succeeding generations, produced by the modification of single base units in DNA, or the deletion, insertion, or rearrangement of larger units of chromosomes or genes. This altering is called a mutation. In this paper, a mathematical model is introduced to this reality. The model describes the time and space for the evolution. The tool is based on a complex domain for the space. We show that the evolution is distributed with the hypergeometric function. The Boundedness of the evolution is imposed by utilizing the Koebe function.     

Coexistence of nearly neutral species

Aims The neutral theory of biodiversity provides a powerful framework for modeling macroecological patterns and interpreting species assemblages. However, there remain several unsolved problems, including the effect of relaxing the assumption of strict neutrality to allow for empirically observed variation in vital rates and the 'problem of time'—empirically measured coexistence times are much shorter than the prediction of the strictly neutral drift model. Here, we develop a nearly neutral model that allows for differential birth and death rates of species. This model provides an approach to study species coexistence away from strict neutrality.Methods Based on Moran's neutral model, which assumes all species in a community have the same competitive ability and have identical birth and death rates, we developed a model that includes birth–death trade-off but excludes speciation. This model describes a wide range of asymmetry from strictly neutral to nearly neutral to far from neutral and is useful for analyzing the effect of drift on species coexistence. Specifically, we analyzed the effects of the birth–death trade-off on the time and probability of species coexistence and quantified the loss of biodiversity (as measured by Simpson's diversity) due to drift by varying species birth and death rates.Important findings We found (i) a birth–death trade-off operating as an equalizing force driven by demographic stochasticity promotes the coexistence of nearly neutral species. Species near demographic trade-offs (i.e. fitness equivalence) can coexist even longer than that predicted by the strictly neutral model; (ii) the effect of birth rates on species coexistence is very similar to that of death rates, but their compensatory effects are not completely symmetric; (iii) ecological drift over time produces a march to fixation. Trade-off-based neutral communities lose diversity more slowly than the strictly neutral community, while non-neutral communities lose diversity much more rapidly; and (iv) nearly neutral systems have substantially shorter time of coexistence than that of neutral systems. This reduced time provides a promising solution to the problem of time.     

Characterization of genes with tissue-specific differential expression patterns in Populus

Like many plants, Populus has an evolutionary history in which several, both recent and more ancient, genome duplication events have occurred and, therefore, constitutes an excellent model system for studying the functional evolution of genes. In the present study, we have focused on the properties of genes with tissue-specific differential expression patterns in poplar. We identified the genes by analyzing digital expression profiles derived by mapping 90,000+ expressed sequence tags (ESTs) from 18 sources to the predicted genes of Populus. Our sequence analysis suggests that tissue-specific differentially expressed genes have less diverged paralogs than average, indicating that gene duplication events is an important event in the pathway leading to this type of expression pattern. The functional analysis showed that genes coding for proteins involved in processes of functional importance for the specific tissue(s) in which they are expressed and genes coding for regulatory or responsive proteins are most common among the differentially expressed genes, demonstrating that the expression differentiation process is under strong selective pressure. Thus, our data supports a model where gene duplication followed by gene specialization or expansion of the regulatory and responsive networks leads to tissue-specific differential expression patterns. We have also searched for clustering of genes with similar expression pattern into gene-expression neighborhoods within the Populus genome. However, we could not detect any major clustering among the analyzed genes with highly specific expression patterns. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible for authorized users.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

一个污染环境中的单种群模型的动力学性质

以脉冲微分方程为基础建立了一个污染环境中在固定时刻对污染净化处理的单种群模型,详细研究了此模型的动力学性质,给出了种群灭绝和持续生存的充分条件.结果表明,当脉冲作用的周期小于某个阈值时,种群将持续生存;否则,种群将趋于灭绝.