### A discrete equivalent of the logistic equation.

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We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator,...

The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation ${\left(r\left(t\right)\Phi \left({y}^{\Delta}\right)\right)}^{\Delta}+p\left(t\right)\Phi \left({y}^{\sigma}\right)=0$, where $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with $\alpha >1$. We discuss sign and smoothness conditions posed on $r$, (non)availability of some transformations, and mainly we show how the behavior of $r$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...