The existence of a positive periodic solution for {dt dP(t) dH(t)=r(t)H(t)[1-K(t) H(t-τ(t)))]-a(t)H(t)P(t), dt dP(t)=-b(t)P(t)+β(t)P(t)H(t)-σ(t)) is established, wherer, K, a, b, βare positive periodic continuous functions with period ω>0, and τ, σare periodic continuous functions with period ω.

Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

In this paper, by using the upper and lower solutions method, we investigate the existence and nonexistence of positive periodic solutions ofin1nite delay functional differential system with a parameter and feedback controls {x(t)=A(t)x(t)-gλF(t,xt,x(g(t,x(t))), u(t)=-B(t)u+E(t,xt,x(h(t,x(t))).

We use the continuation theorem of coincidence degree theory and Liapunov functions to study the existence and stability of periodic solutions for the Cohen-Grossberg neural network with multiple delays.

By using the continuation theorem of coincidence degree theory, sufficient and realistic conditions are obtained for the existence of positive periodic solutions for the periodic distributed delay Lotka-Volterra competition system and the periodic state dependent delay Lotka-Volterra competition system dt dui(t) ui(t)[ri(t) aii(t)ui(t) n∑j1ji aij(t)∫oTij Kij(s)uj(t s)ds],i 1, 2……, n, and the periodic state dependent delay Lotka-Volterra competition system dt dui(t) ui(t)[ri(t) aij(t)ui(t) n∑j1ji aij(t)uj(t τj(t,u1(t),……,un(t)))],i 1, 2,……, n, where ri, aii>0, aij 0(j i, i, j 1, 2, …, n)are continuous ω-periodic functions, Tions, Tij (0,∞)(j i, i, j 1, 2, …, n), Kij C(tij, 0, (0,∞)), ∫0 Tij Kij (s) ds 1(j i, i, j 1, 2, …,n), τi C(Rn 1, R), and τi(i 1, 2, …, n) are ω-periodic with respect to their first arguments, respectively.