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In this paper, we develop a general approach to the existence of periodic solutions for the difference equation[image omitted] with variable delays and apply it to find necessary and sufficient conditions for the existence of a positive periodic solution [image omitted] for the host macroparasite model. We present some sufficient conditions for the global attractivity of [image omitted] . For the constant coefficients nondelay case, we obtain necessary and sufficient attractivity conditions using recent results of Liz. [ABSTRACT FROM AUTHOR]

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation[image omitted] on a time scale  where γ ≥ 1. As a special case when  =  our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233-240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123-125) but also improve these results. When  = ,  = h or  = qN, i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples. [ABSTRACT FROM AUTHOR]

We consider the second part of the Cushing-Henson conjecture (the cycle's average is less than the average of carrying capacities; the first part of the conjecture deals with the existence and global stability of periodic cycles) for a periodic delay difference equation [image omitted] Sufficient conditions on f and hi are obtained, when the second part of the conjecture is valid. We demonstrate the sharpness of these conditions by presenting several counterexamples. In addition, sufficient global attractivity conditions are deduced for the Pielou equation. [ABSTRACT FROM AUTHOR]