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The aim of this article is to study the dynamical behavior of a Lotka-volterra 3-species ratio-dependent predator-prey system with delays and feedback controls. By using the comparison theorem, the differential inequalities and developing new analysis method, some sufficient conditions are obtained to ensure the permanence of the solutions for the delayed predator-prey system, and some known results are generalized.

The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation: xn+1=Axn−1xn−2/B+∑i=3kCixn−i n=0,1,2,…, where xn is the sequence of positive fuzzy numbers and the parameters A,B,C3,C4,…,Ck and initial conditions x0,x−1,x−2,x−ii=3,4,…,k are positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.

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