Permancence And Global Asymptotic Stability In Three-species Non-autonomous Systems With Time-delay

In this article, we will study the permanence and global asymptotic stability of the biological community. Firstly, several models about the biological community are given, in-cluding population competition, mutual benefit and predator-prey system. In addition, it also gives some reaction functions, such as Holling functional reaction and ratio-dependent func-tion. In this article, we mainly study the following three-species non-autonomous predator-prey systems with time-delay.The initial conditions of (1) are given asx1(θ)=φ1(θ),x2(θ)=φ2(e),x3(θ)=φ3(0), (τ<θ<0) (2)Here, we have gained some sufficient conditions on permanence of the system by using inequality analytical technique. By Lyapunov functional methods, we have also established some conditions for global asymptotic stability. The main conclusions are following.定理1For anyφ∈C+, the system's result in[0,+∞] is only one, and for any?≥0. the result xi(t)>0 (i=1,2,3).定理2 if,then the system(1) is persistent.定理3If there existα1> 0,α2>0,α3>0, and nonncgative function a(t), +∞satisfies max[-B1(t),-B2(t),-B3(t)}≤-α(t)Then the system(1)is global asymptotic stability. Where where m, Mare given in theorem 2.2In addition, we make some changes in the system above, we can get another one:By similar methods, we can get sufficient conditions on the permanence and global asymp-totic stability of this system.