Discrete random walk model to interpret the dispersal parameters of organisms

Most models for describing the dispersal of organisms have been developed by using diffusion equations with a diffusion coefficient. These equations, however, do not yield information that is readily interpretable in biological terms, because the biological meaning of the diffusion coefficient is not always clear. Discrete random walk models, in which organisms move into adjacent positions by a specific probability, seem to be superior in their biological tractability, although they have not been as widely used as diffusion equations because of their mathematical intractability. We reconstructed discrete random walk models of one dimension based on two assumptions: (1) moving organisms settle by a constant probability and (2) settled individuals are captured by traps by a constant probability. We also constructed a model that is applicable for a directional movement caused by environmental factors such as wind. We applied the model to a one-dimensional dispersal experiment on the ragweed beetle, Ophraella communa LeSage, an insect of the size of about 4 mm in adults. Both larvae and adults of this species preferably eat ragweed, Ambrosia artemisiifolia L. We planted ragweed plants at a place in a linear field of 100 m length and 20 m width. In mid-August, adult beetles dispersed actively along the linear field to find new food plants after they almost defoliated ragweed plants. Assuming a non-directional random walk, we performed the linear regression, which indicated that the movement of the adult O. communa in this season is approximately described by a discrete random movement in which an individual travels next 10 m with a probability of 0.906 during its life. The dispersion parameter estimated by the Poisson regression was much larger than 1, indicating that there is a considerable amount of fluctuation in the probability of capture or in other parameters. It is shown that a similar model is also applicable to a situation in which individuals are directly removed by traps from the moving population before settlement.