In this paper we incorporate the white noise and Markovian switching into the modified Leslie-type predator-prey model with foraging arena scheme. Especially, the parametric conditions for the system to be permanent or extinct are investigated. Inspired by \cite{Dieu2016classification,hening2017stochastic}, we also consider how the correlations between the Brownian motions affect the population interactions. Numerical simulations are carried out to substantiate the analytical results.