Theoretical Study of the Functional Differential Equations using Computer Simulation

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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