In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier–Stokes equations in Ω ⊂ R 2 . The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data ( ρ 0 , u 0 ) ∈ L ∞ × H 0 s with s > 0, which constitutes a positive answer to the question raised by Danchin and Mucha (2019 Commun. Pure Appl. Math. 72 1351–85), in which the initial velocity u 0 ∈ H 0 1 (see also Li (2017 J. Differ. Equ. 263 6512–36).

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.