Theoretical and Numerical Study of the Functional Differential Equations

主要研究成果如下：1.改进了本课题组几年前开发的一维和二维多时滞微分方程组的计算机模拟软件DDEIRK和DDE2RK。现在，我们得到了适用于积分时滞微分方程的新的计算软件FDEIRK和FDE2RK。我们还开发了高阶积分项时滞微分方程解的计算机模拟软件FDE4RKP和FDE4RKT。利用这些软件FDE4RKP和FDE4RKT，我们研究了数学生态中时滞微分方程解的性态，发现了解的一些有趣的性质。我们成功地给出了这些性质的数学证明。其典型结果如下：(1)n维变系数线性泛函微分方程零解指数渐近稳定的充要条件；(2)一维变系数非线性泛函微分方程零解渐近稳定的最佳可能充分条件；(3)一类Lotka-Volterra型时滞微分方程全局渐近稳定和持久生存的充要条件；(4)以积分形式表示的捕食者-食饵方程渐近稳定的充分条件。用Stieltjes积分表示的时滞线性泛函微分方程解的渐近稳定性和渐近常数问题。本文在数学生态学中找到了无时滞的食饵-捕食者方程存在极限环的充要条件，并成功地给出了这些定理的数学证明。

Main results of our research are as follows :1. We have improved computer softwares DDEIRK and DDE2RK which were developed in our group several years ago for computer simulation of one and two dimensional differential equations with several delays. Now we obtained new computer softwares FDEIRK and FDE2RK which are applicable to delay differential equations with integral terms. We also developed computer softwares FDE4RKP and FDE4RKT for computer simulation of soluions of higher order dimensional delay differential equations with integral terms.2. Using these softwares FDE4RKP and FDE4RKT, we studied the behavior of solutions of delay differential equations in mathematical ecology and found some interesting properties of solutions. We succeeded to give mathematical proofs to these properties. The typical results are as follows :(1) A neccesary and sufficient condition for the exponential asymptotic stability of the zero solution of n-dimenssional linear functional differential equations with variable coefficients.(2) A best possible sufficient condition for the asymptotic stability of the zero solution of one dimen- sional nonlinear functional differential equations with variable coefficient.(3) Neccesary and sufficient conditions for the global asymptotic stability and the permanence of a Lotka-Volterra type delay differential equations(4) Sufficient conditions for the asymptotic stability of a prey-predator differential equation with delays expressed in integral terms.(5) Neccesary and sufficient conditions for the asymptotic stability and the asymptotic constant problem of a linear functional differential equation with delays expressed in Stieltjes integral.3. We found neccesary and sufficient conditions for the existence of a limit cycle of prey-predator differential equations without delay of Ivlev type and Holling type in mathematical ecology and succeeded to give mathematical proofs to these theorems.