Theoretical and Numerical Study of the Functional Differential Equations

点击翻译按钮获取中文摘要

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.