For integrated pest management (IPM), we propose a generalized stage-structured plant-pest-natural enemy system with impulsive spraying pesticide and releasing natural enemies at different fixed moment. By the stroboscopic maps, we obtain two types of periodic solutions: the plant-pest-extinction and the pest-extinction. The sufficient conditions for the global attractivity of a pest-extinction periodic solution and permanence of the system are obtained by comparison theorem and stroboscopic technique. Moreover, numerical simulations are inserted to verify the effectiveness and feasibility of the theoretical results, which show that the impulsive control plays a key role on the permanence of the system.

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in ${\mathbb{R}}^d(d\ge 3)$. And the global well-posedness and Gevrey analytic of strong solutions for the system are established in the $L^2-L^p$ type critical hybrid Besov space with $2\le p\le \frac{2d}{d-2}$ and $p<d$.

In this note, we study the Cauchy problem of the fractional Fokker-Planck equation. We prove that the solution to the Cauchy problem enjoys the analytic smoothing effect with an L 2 initial datum for positive time, and the evolution of the analytic radius is similar to the heat equation.