The authors study the n-th order nonlinear neutral differential equations with the quasi – derivatives $L_n[x\left(t\right)+{(-1)}^{r}P\left(t\right)x\left(g\left(t\right)\right)]+\delta Q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)=0,$ where $n\ge 2,r\in \{1,2\},$ and $\delta =\pm 1.$ There are given sufficient conditions for solutions to be either oscillatory or they converge to zero.

This paper deals with the second order nonlinear neutral differential inequalities $\left(A_{\nu }\right)$: ${(-1)}^{\nu }x\left(t\right)\{z^{\text{'}\text{'}}\left(t\right)+{(-1)}^{\nu }q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)\}\le 0,$$t\ge t_0\ge 0...$

We present several results on permanence and global exponential stability of Nicholson-type delay systems, which correct and generalize some recent results of Berezansky, Idels and Troib [Nonlinear Anal. Real World Appl. 12 (2011), 436-445].

We study a generalized Nicholson's blowflies model with a nonlinear density-dependent mortality term. Under appropriate conditions, we employ a novel proof to establish some criteria guaranteeing the permanence of this model. Moreover, we give an example to illustrate our main result.

The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.

We study solutions tending to nonzero constants for the third order differential equation with the damping term $${\left(a_1\left(t\right){\left(a_2\left(t\right)x^{\text{'}}\left(t\right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right)f\left(x\left(\varphi \left(t\right)\right)\right)=0$$ in the case when the corresponding second order differential equation is oscillatory.

The paper considers a scalar differential equation of an advance-delay type $$\dot{y}\left(t\right)=-\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma ),$$ where constants $a_0$, $b_0$, $\tau$ and $\sigma$ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\to \infty$ is analyzed provided that the transcendental equation $$\lambda =-a_0{\mathrm{e}}^{-\lambda \tau }+b_0{\mathrm{e}}^{\lambda \sigma }$$ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.