关于Lotka-Volterra型积分微分方程解的渐近状态

On the Asymptotic Behavior of Solutions for Lotka-Volterra Type Integro-Differential Equations

本文考虑形如(t)=diag(x(t)){diag(b)x(t)+(integral from n=-r to 0([dμ(s)]x(t+s)}的Lotka-Volterra型积分微分方程,给出了这类方程的所有正解收敛于零或者发散到无穷的充分条件.

The Lotka-Volterra type integro-differential equations of the form-x(t)=diag(x(t)){ diag(6)x(t) + [du(s)]x(t + s)} are studied, where b Rn,r≥0 and u is an Rnxn-valued measure on [-r,o]. The main results show that the asymptotic behavior of the positive solutions for such an equa-tion is determined by the stability modules s(B) of B=diag(b) + du. Tobe exact, if u≥0, s(B)≤0, all positive solutions for the equation will converge to zero as t approaches infinity; if u≥0, s(B)=0 and B is irreducible, then all positive solutions for the equation are defined on [0,(∞)) and bounded; and if u≥0, s(B)>0 and B is irreducible, then all positive solutions for the equation are unbounded. The results derived imply the-main theorems given by G. Karakostas and I. Gyori.