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We propose an impulsive integro-differential model to describe the dynamics of an invading species with a pulsive reproduction stage and a nonlocal dispersal stage. We first establish a threshold-type result on the global dynamics of the system in a bounded domain, and present an application to insect pests outbreak in terms of the critical domain size. For the spatial spread in an unbounded domain, we then prove the existence of the invasion speed and its coincidence with the minimal speed for monotone traveling waves. Numerical simulations are also carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

West Nile virus (WNv) is a mosquito-borne disease caused by Flavivirus. To investigate the combined effects of vertical transmission, temperature-dependent incubation periods and seasonality on the transmission of WNv, we develop a delay differential system with stage-structure and time-varying delays. We then derive the mosquito reproduction number R0V and basic reproduction number R0 and show that these two numbers serve as threshold parameters that determine whether WNv will spread. As an application, we conduct a case study for WNv transmission in Los Angeles County, California. We also carry out numerical simulations to identify the situations that require time-periodic delays. Moreover, we find that rising temperatures may potentially increase the risk of disease outbreaks. [ABSTRACT FROM AUTHOR]