Fly communities exhibit rich ecological dynamics, and one of the important influencing factors is the interaction between species. A discrete Nicholson-type system with multiple time varying delays which considers the mutualism relationship between two fly species is investigated in this paper. Sufficient conditions for the existence of positive periodic solutions are elucidated. The result is obtained by the well-known continuation theorem of coincidence degree theory. An example is attached to illustrate our result. Moreover, the actual biological descriptions obtained from our main result are explained.

In this paper, we investigate the multiplicity of positive periodic solutions of a discrete blood cell production model with impulse effects. This model is described by periodic coefficients and time delays, as well as nonlinear feedback with exponential terms. By employing the Krasnosel'skii fixed point theorem, we establish a sufficient condition for the existence of at least two positive periodic solutions. To this end, we construct solution transformation between an impulsive delay difference equation and the corresponding nonimpulsive delay difference equation. Aditionally, a solution representation of the positive periodic solution of the blood cell production model is presented. Moreover, a numerical example and its simulations are given to illustrate the main result.

In this paper, we propose a detrital-based mangrove food chain system with Holling type Ⅱ functional response, intraspecific competition, and delay effects. First, we prove the solution of this system is positive and bounded with the positive initial conditions. Second, we calculate the equilibria and investigate the asymptotical stability of equilibria with and without delays. Then, by taking the delay as the bifurcation parameter and using bifurcation theory, the bifurcation conditions for the system to undergo Hopf bifurcation at the interior equilibrium point are obtained. Furthermore, we also conduct the length estimation of delay to preserve the stability by using the Nyquist criterion. Finally, with the suitable choices of the parameters, numerical simulations have been carried out to substantiate our analytical results.