Huang et al. developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers (see SIAM Journal on Applied Mathematics, 77 (2017), 854-880). 

They used a net reproductive rate R0 to determine population persistence in a bounded domain and estimated spreading speeds in a unbounded domain. 

Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. 

In this work, we analyze the spatial dynamics of this model mathematically.

We first solve the parabolic equation and rewrite the model into a fully discrete-time model. 

In a bounded domain, we show that the spectral radius of the linearized operator can be used to determine population persistence and also confirm that R0 defined Huang et al's work can be used to determine population persistence.

In an unbounded domain, we construct two monotonic operators to control the model operator from above and from below and obtain upper and lower bounds of the spreading speeds of the model.