From individual behaviour to population models: a case study using swimming algae

Trajectories of swimming algae are analysed, and two random-walk models developed to link the individual-level behaviour of cells to population-level advection-diffusion models for the spatial-temporal distribution of cells. The models are both of the advection-diffusion form but are based on two different sets of assumptions about the underlying random-walk behaviours, a velocity jump behaviour and a velocity diffusion behaviour. The mathematical models were developed to allow for an arbitrary (non-weak) bias in the random walk and a variable swimming speed in order to represent the experimental data. For the algal species considered, Heterosigma akashiwo, the mean upward swimming speed was computed as 40 microm s(-1), and the calculated diffusion constants ranged from 2 x 10(3) to 4 x 10(4) microm(2) s(-1) depending on the details of the models. That two widely used modelling approaches yield substantially different population-level predictions when applied to the same empirical data suggests that better theoretical tools are needed for identifying adequate approximations for behavioural characteristics.