In this survey, we briefly review some of our recent studies on predator-prey models with discrete delay. We first study the distribution of zeros of a second degree transcendental polynomial. Then we apply the general results on the distribution of zeros of the second degree transcendental polynomial to various predator-prey models with discrete delay, including Kolmogorov-type predator-prey models, generalized Gause-type predator-prey models with harvesting, etc. Bogdanov-Takens bifurcations...

Qualitative comparison of the nonoscillatory behavior of the equations $$L_{n}y\left(t\right)+H(t,y\left(t\right))=0$$ and $$L_{n}y\left(t\right)+H(t,y\left(g\left(t\right)\right))=0$$ is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form $$L_n=\frac{1}{p_n\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p_{n-1}\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}...\frac{\mathrm{d}}{\mathrm{d}t}\frac{·}{p_0\left(t\right)}.$$ Both canonical and noncanonical forms of $L_n$ have been studied.

We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations $$u^{\text{'}\text{'}}\left(x\right)+\sum _i{p}_i\left(x\right)u^{\text{'}}\left(h_i\left(x\right)\right)+\sum _i{q}_i\left(x\right)u\left(g_i\left(x\right)\right)=0$$ without the delay conditions $h_i\left(x\right),g_i\left(x\right)\le x$, $i=1,2,...$, and $$u^{\text{'}\text{'}}\left(x\right)+{\int }_0^{\infty }{u}^{\text{'}}\left(s\right){\mathrm{d}}_s{r}_1(x,s)+{\int }_0^{\infty }u\left(s\right){\mathrm{d}}_s{r}_0(x,s)=0.$$

In this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form $${\left(r_2\left(t\right){\left(r_1\left(t\right)x^{\text{'}}\right)}^{\text{'}}\right)}^{\text{'}}+p\left(t\right)x^{\text{'}}+q\left(t\right)f\left(x\left(g\left(t\right)\right)\right)=0.$$ We establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero.

We study oscillatory properties of solutions of the systems of differential equations of neutral type.