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With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi-ratio-dependent predator–prey system with time delays and impulses, which covers many models appeared in the literature. When the results reduce to the semi-ratio-dependent predator–prey system without impulses, they generalize and improve some known ones.

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions in delayed Gause-type predator–prey systems. Some known results are shown to be special cases of the presented paper.

An n species nonautonomous competitive Lotka–Volterra system is considered in this paper. The average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. The generalization for the result is presented, that is, for each r ⩽ n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n − r are driven to extinction. It is shown that these average conditions are improvement of those of Ahmad and Montes de Oca [Appl. Math. Comput. 90 (1998) 155–166] and Montes de Oca and Zeeman [Proc. Amer. Math. Soc. 124 (1996) 3677–3687] and [J. Math. Anal. Appl. 192 (1995) 360–370].

This paper investigates the qualitative behavior of solutions of the autonomous planar system with zero diagonal coefficient x =p 2 (y)q 2 (x)y , y =p 3 (y)q 3 (x)x+p 4 (y)q 4 (x)y . Under suitable assumptions, the necessary and sufficient conditions for all solutions to be oscillatory, and for the origin to be a global center are established. The theorems on the existence and uniqueness of nontrivial periodic solutions are also proved.

In this paper, we use the theory of generalized rotated fields to prove a theorem of the uniqueness of limit cycle for the generalized Lienard system. The damping terms of this system are dependent on x . Our result extends a famous Zhang–Zhifen's theorem on the uniqueness problem of limit cycle, which is contributed the Lienard systems whose damping terms are independent of x .