延迟遗传调控网络稳定性及分支

本文主要研究了延迟遗传调控网络的局部稳定性和该网络的Hopf分支存在条件．延迟遗传调控网络是无穷维系统,此类系统在平衡点线性化后的特征方程为超越方程。通过对此超越方程进行研究,得到了系统系数不同时的系统稳定的条件及相关结论,又进一步说明了此系统的Hopf分支存在条件．最后,举一个例子进行了数值仿真验证了所得到的结论．     

食饵具有周期强迫的捕食模型

建立并研究了一类具有周期强迫和脉冲扰动的捕食模型,通过理论分析和数值模拟,得到了食饵灭绝周期解全局渐近稳定和系统持久的充分条件,利用分支理论证明了边界周期解附近会分支出正周期解.     

具收获与投放的三种群Volterra模型的周期轨道

本文在文〔2〕的基础上,研究三维动力系统的Lyapunov分枝问题．放宽了文〔2〕中关于向量场函数的限制,相应地给出从空间闭轨族扰动产生孤立周期解的判定方法．应用判定定理及椭圆函数的积分技巧,研究了具有收获与投放的三种群Volterra模型,得到了存在周期轨道的充分条件．     

森林发展系统强解的存在唯一性及稳定性分析

对森林发展系统的动态数学模型,运用泛函分析和积分方程的理论,证明了系统强解的存在性和唯一性,并对系统的稳定性进行了讨论．     

具HollingⅡ类功能反应的多时滞捕食-食饵系统的全局性质

本文研究了一类具HollingⅡ功能反应的捕食-食饵系统,首先用Cook等人建立的关于超越函数的零点分布定理,研究了一类多时滞捕食-食饵系统的正平衡点的稳定性及局部Hopf分支．进而,再结合吴建宏等人的用等变拓扑度理论建立起的一般泛函微分方程的全局Hopf分支定理,进一步研究了该系统的全局Hopf分支的存在性．     

具脉冲出生的Leslie-Gower捕食者-食饵系统的动力学分析

研究了一个具有脉冲出生的Leslie-Gower捕食者一食饵系统的动力学性质．利用频闪映射。得到了带有Ricker和Beverton-Holt函数的脉冲系统准确的周期解．通过Floquet定理和脉冲比较定理,讨论了该系统的灭绝和持久生存．最后,数值分析了以b（p）为分支参数的分支图,得到的结论是脉冲出生会带给系统倍周期分支、混沌以及在混沌带中出现周期窗口等复杂的动力学行为．     

中国独活属的分支分类学研究

研究运用分支分析方法研究了16种独活属植物的系统发育关系。采用最简约性分析的分支与界限法Branch—and—Bound(bandh)的严格一致树(thestrict consensus cladogram)和非加权配对算术平均法up weighted pair group method using arithmetic average method(UPGMA)对其进行分支分析。研究结果表明：两种方法均支持将独活属分为4个组,即多裂叶组Sect．Millefolia、多毛组Sect．Villosa、多管组Sect．Plurivittata、独活组Sect．Heracleum。《中国植物志》中少管组Sect．Wendia的法落海Heracleum apaense没有独立成组,它的系统位置还需要进一步研究。根据PAUP软件分析中的不同处理方法,探讨了中国独活属植物的系统进化及其起源问题。     

描述神经元相互作用的非线性动力学方程的解析解

神经系统是由大量神经元构成的非线性动力系统,动力学行为十分复杂ＦＨＮ模型提供了在该系统中观察时间周期振荡这种非线性现象的实验证据．本文利用扰动法从理论上求出了ＦＨＮ模型所给出的非线性动力学方程的解析解,为进一步深入研究神经系统的动力学行为提供了理论依据     

具有变消耗率微生物连续培养模型的定性分析

研究了一类具有变消耗率的微生物连续培养系统,当消耗率是线性函数时得到了正平衡点全局渐近稳定的充要条件,当消耗率是二次函数时得到了系统存在极限环的充分条件,同时利用分支理论研究系统存在Hopf分支的条件,判定了极限环的稳定性．     

Banach空间二阶非线性微分—积分方程两点边值问题整体解的存在性

利用一个新的比较结果和Moench不动点定理，研究了Banach空间二阶非线性微分－积分方程两点边值问题整体解的存在性。     

A deconvolution scheme

A deconvolution scheme is presented for the calculation of an input function given a system and its response—by converting the integral equation into an initial value problem. This technique is particularly simple and advantageous when dealing with compartmental systems and solving them numerically. The method also permits the imbedding of constraints and the estimation of confidence limits for the deconvoluted function.     

Othenio Abel (1875–1946) and “the phylogeny of the parts”1

The reconstruction of the evolutionary history of animal phyla was an integral part of Othenio Abel’s paleobiology (paleozoology). Abel took issue with those phylogeneticists who, following the lead of Haeckel, would draw up phylogenetic trees on the basis of transformation series of singular characters considered to be of particular importance. Abel highlighted Louis Dollo’s principle of the chevauchement des spécialisations (crossing of specializations), which transformed phylogenetics from a search for ancestor–descendant sequences to research into relative degrees of relationships. This replacement resolved the conflict, much discussed at the time, between the continuity of ancestor–descendant lineages and the discontinuity inherent in the natural (phylogenetic) system. Walter Zimmermann refined Abel’s methodology, which he called character‐phylogenetics (Merkmalsphylogenie), an approach that was eventually adopted by Willi Hennig.     

Modelling of the control of heart rate by breathing using a kernel method

The process of the breathing (input) to the heart rate (output) of man is considered for system identification by the input-output relationship, using a mathematical model expressed as integral equations. The integral equation is considered and fixed so that the identification method reduces to the determination of the values within the integral, called kernels, resulting in an integral equation whose input-output behaviour is nearly identical to that of the system. This paper uses an algorithm of kernel identification of the Volterra series which greatly reduces the computational burden and eliminates the restriction of using white Gaussian input as a test signal. A second-order model is the most appropriate for a good estimate of the system dynamics. The model contains the linear part (first-order kernel) and quadratic part (second-order kernel) in parallel, and so allows for the possibility of separation between the linear and non-linear elements of the process. The response of the linear term exhibits the oscillatory input and underdamped nature of the system. The application of breathing as input to the system produces an oscillatory term which may be attributed to the nature of sinus node of the heart being sensitive to the modulating signal the breathing wave. The negative-on diagonal seems to cause the dynamic asymmetry of the total response of the system which opposes the oscillatory nature of the first kernel related to the restraining force present in the respiratory heart rate system. The presence of the positive-off diagonal of the second-order kernel of respiratory control of heart rate is an indication of an escape-like phenomenon in the system.     

Bionomics dynamic model of a class of competition systems ☆

Differential equation problem is an important research topic in the international academia. In accordance with certain ecological phenomena, previous research was conducted based on simple observational and statistical data. But this approach does not effectively study the essence of the ecological phenomena. Recently, one dynamic approach has been proposed for the study of ecology in the international academia. According to this approach, first of all, the ecology is reduced to the differential equation model which represents the essential phenomenon, and then the dynamic law and rules of mathematics and biology will be studied. Currently, an extensive research is conducted on the differential equation problem. This paper primarily explores a type of competitive ecological model, which is a system of differential equation with infinite integral. we first study the existence of positive periodic solution to this model, and then present sufficient conditions for the global attractivity of positive periodic solutions.     

Plasma outflow from a corrugated trap in the kinetic regime

The problem of stationary plasma outflow from an open corrugated trap in the kinetic regime is considered with allowance for pair collisions in the framework of a kinetic equation with the Landau collision integral. The distribution function is studied in the limit of small-scale corrugation and a large mirror ratio. In considering a single corrugation cell, a correction for the distribution function is calculated analytically. An equation describing variations of the distribution function along the system is derived and used to study the problem of plasma outflow into vacuum.     

A parameter identification problem arising from a model of canalicular bile formation

We develop a simple mathematical model for bile formation and analyze some features of the model that suggest the design for future physiological experiments. The mathematical model results in a boundary value problem for a system of functional differential equations depending on several physical parameters. From the observability of the boundary values we can identify, both qualitatively and quantitatively, some of these physical parameters. This identification then suggests physical experiments from which one could infer some of the bile transport phenomena that are not, at present, directly observable. The mathematical parameter identification problem is solved by converting the boundary value problem to a transition time problem for a quadratic system of ordinary differential equations on the plane where we are able to employ some special properties of quadratic systems in order to obtain a solution.The author was supported by the Air Force Office of Scientific Research and the National Science Foundation under the grants AF-AFOSR-89-0078 and DMS-9022621The author was supported by National Institutes of Health under grant number R37 DK-27623     

Mathematical analysis of an integral equation arising from population dynamics

In this paper, we establish the existence of travelling wave solution to an intrinsically non-linear differential–integral equation formed as a result of mathematical modelling of the evolution of an asexual population in a changing environment. This equation is first converted to a non-linear integral equation. The discretization and manipulation of the corresponding eigenvalue problem allows us to use the theory of positive matrices to get some very useful estimates and then to confirm the existence of solution. We also exhibit numerical simulation results and explain the biological meaning of the results.     

Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible,partly injectable multicompartment systems

An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.     

Transient behavior of the single loop solute cycling countercurrent multiplier

Transient behavior of a single loop solute cycling countercurrent multiplier is described by a Volterra type integral equation similar to that describing circulation of an indicator in the systemic circulation. Solution of the equation is given for pumping from ascending to descending flow proportional to concentration in ascending flow and no back leak. This exact solution is compared with an approximate solution of the integral equation and with a solution in which the flow system is represented by a finite Markov chain. Agreement between the Markov approximation and the exact solution is excellent.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.