Problem SE6

The zebra mussel (Dreissena polymorpha, Fig. 3) is a freshwater invasive species which in the past 25 years has been responsible for enormous damages, both environmental and economic, specially in the United States (and marginally in Italy too). The average lifetime of D. polymorpha is about 4 years, and each female produces in the average 2 $\times 10^5$ eggs per year, with sex ratio 1:1. The fraction of eggs surviving up to adulthood is very low, approximately equal to 0.005%. Adult individuals are basically sessile, while larvae can be transported by the river stream for a few days, after which they settle in the river or lake bed and become mature adults that can reproduce.

Recently, the species has been accidentally introduced into the Yellowish River, downstream of the dam of Deep Lake. The downstream expansion speed of the species has been estimated to be 640 km/year. You know that the river current implies a drift of the larvae of about 600 km/year. Estimate the diffusion coefficient of D. polymorpha, assuming that the spatiotemporal dynamics of the population is well described by a model including advection, diffusion and Malthusian growth.

Figure 3: Zebra mussels.