### A general discrete time model of population dynamics in the presence of an infection.

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A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback...

The asymptotic behaviour for solutions of a difference equation ${z}_{n}=f(n,{z}_{n})$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.

In the paper we consider the difference equation of neutral type $${\Delta}^{3}[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\in \mathbb{N}\left({n}_{0}\right),$$ where $p,q:\mathbb{N}\left({n}_{0}\right)\to {\mathbb{R}}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma $ is strictly increasing and $\underset{n\to \infty}{lim}\sigma \left(n\right)=\infty ;$$\tau $ is nondecreasing and $\underset{n\to \infty}{lim}\tau \left(n\right)=\infty $, $f:\mathbb{R}\to \mathbb{R}$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda}^{*}<1,\phantom{\rule{1.0em}{0ex}}\sigma \left(n\right)=n-k,\phantom{\rule{1.0em}{0ex}}\tau \left(n\right)=n-l,$$ and $$1<{\lambda}_{*}\le p\left(n\right),\phantom{\rule{1.0em}{0ex}}\sigma \left(n\right)=n+k,\phantom{\rule{1.0em}{0ex}}\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty $ with a weaker assumption on $q$ than the...

Asymptotic properties of solutions of the difference equation of the form $${\Delta}^{m}{x}_{n}={a}_{n}\varphi ({x}_{{\tau}_{1}\left(n\right)},\cdots ,{x}_{{\tau}_{k}\left(n\right)})+{b}_{n}$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta}^{m}{y}_{n}={b}_{n}$ is asymptotically equivalent to some solution of the above equation are obtained.

Asymptotic properties of solutions of difference equation of the form $${\Delta}^{m}{x}_{n}={a}_{n}{\varphi}_{n}\left({x}_{\sigma \left(n\right)}\right)+{b}_{n}$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta}^{m}{y}_{n}={b}_{n}$ is asymptotically equivalent to some solution of the above equation are obtained. Moreover, the conditions under which every polynomial sequence of degree less than $m$ is asymptotically equivalent to some solution of the equation and every solution is asymptotically polynomial are obtained. The consequences of the existence of asymptotically...