A Hybrid Continuous/Discrete-Time Model for Invasion Dynamics of Zebra Mussels in Rivers

While some species spread upstream in river environments, not all invasive species are successful in spreading upriver. Here the dynamics of unidirectional water flow found in rivers can play a role in determining invasion success. We develop a continuous-discrete hybrid benthic-drift population model to describe the dynamics of invasive freshwater mussels in rivers. In the model, a reaction-advection-diffusion equation coupled to an ordinary differential equation describes the larval dispersal in the drift until settling to the benthos, while two difference equations describe the population growth on the benthos. We study the population persistence criteria based on three related measures: fundamental niche, source-sink distribution, and net reproductive rate. We calculate the critical domain size in a bounded domain by analyzing a next generation operator. We analyze the upstream and downstream spreading speeds in an unbounded domain. The model is parameterized by available data in the literature. Combining the results of model parameterization and theoretical analysis, we numerically analyze how the interaction between population growth and dispersal, river flow rate, and water temperature affects both persistence and the spread of zebra mussels along a river.