Theoretical Study of the Functional Differential Equations using Computer Simulation

主要研究成果如下：(1)对本课题组两年前开发的用于二维积分项微分方程计算机模拟的计算机软件FDE2RK进行了改进。在积分核不依赖于时间变量的情况下，我们改进了我们的软件，可以计算我们的微分（II）。利用FDE2RK软件，我们研究了数学生态学中时滞微分方程解的行为，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。典型结果如下：(1)一类具有双时滞的Lotka-Volterra型微分方程全局渐近稳定的充分必要条件(2)一类具有分布时滞的SIR流行病模型的稳定性分析(3)一类具有时滞的二维神经网络的稳定性分析（III）得到了一类不具有时滞的捕食-捕食微分方程存在极限环和全局渐近稳定的充分必要条件并成功地对这些定理进行了数学证明。(5)利用DDE2E软件，研究了带分段常参数的时滞微分方程和带时滞的差分方程解的行为，发现了解的一些有趣性质。我们成功地给出了这些性质的数学证明。典型的结果有：(1)高阶线性微分方程全局渐近稳定的充分必要条件(2)Lotka-Volterra型差分方程持久的充分必要条件(3)分段常参数二维线性微分方程的动力学分类

Main results of our research are as follows :(I) We have improved computer software FDE2RK which was developed in our group two years ago for computer simulation of two dimensional differential equations with integral terms. In case the kernel of the integral does not depend on time variable, we improved our software and we can compute our differential(II) Using the software FDE2RK, 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) Neccesary and sufficient conditions for the global asymptotic stability and the permanence of a Lotka-Volterra type differential equations with two delays(2) Siability analysis of SIR epidemic models with distributed delays(3) Stability analysis of two-dimensional neural net work with delays(III)We found neccesary and sufficient conditions for the existence of a limit cycle and the global asymptotic stability of prey-predator differential equations without delay of Holling type in mathematical ecology and succeeded to give mathematical proofs to these theorems.(VI) Oscillation theorems of the Riemann-Weber version of Euler differential equations with delay(V) Using the software DDE2E, we studied the behavior of solutions of delay differential equations with piecewise constant arguments and difference equations with delays and found some interesting properties of solutions. We succeeded to give mathematical proofs to these properties. The typical results are as follows :(1) Neccesary and sufficient conditions for the global asymptotic stability of a linear defference equation of higher order(2) Neccesary and sufficient condition for the permanence of difference equations of Lotka-Volterra type(3) Classification of the dynamics of 2-dimensional linear differential Equations with piecewise constant arguments