The spatial dynamics of a Zebra mussel model in river environments

Abstract

Huang et al. [10] developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers. They used a net reproductive rate $ R_0 $ to determine population persistence in a bounded domain and estimated spreading speeds by applying the linear determinacy conjecture and using the formula in [16]. Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. In this paper, 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 $ \hat{r} $ of the linearized operator can be used to determine population persistence and that the sign of $ \hat{r}-1 $ is the same as that of $ R_0-1 $, which confirms that $ R_0 $ defined in [10] 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. Erratum: The name of the second author has been corrected from Xiang-Qiang Zhao to Xiao-Qiang Zhao. We apologize for any inconvenience this may cause.