Encounter rate problems arise frequently in ecology and many other scientific disciplines. Many encounter problems can be framed such that a moving “searcher” encounters some stationary “target” if it comes within a specified detection radius of the target. Ecological examples include individuals searching for food or other resources, a mate, or in less fortunate cases, a sit-and-wait predator or a passive sampling device such as a pitfall trap. Available models for this type of encounter problem presently come in two forms: ballistic and diffusive. Ballistic models assume that searchers move in straight lines oriented randomly in space, while diffusive encounter models assume that searchers trace infinitely wiggly paths. These two end points of the movement behavior spectrum are both analytically tractable and well studied. However, when movement is by correlated random walk and the average distance over which directional persistence is maintained—the correlation length scale of the random walk—is greater than the target detection radius, neither of these models accurately predicts the mean number of unique encounters in a given time period. Under these conditions, the encounter process itself is ballistic, but the classical ballistic encounter model fails to account for the turning component of the random walk.
Results/Conclusions
Focusing on empirical ground beetle pitfall trapping studies, we show that situations where the correlation length scales of searchers’ random walks exceed their target detection radii may often occur in real-world encounter problems. We then provide a heuristic derivation of an encounter model for correlated random walks with ballistic encounter that can be expressed in terms of random walk parameters, searcher population density, and target detection radius. Using extensive numerical simulations, we demonstrate that the new encounter model is quantitatively accurate across a wide range of relevant scenarios, and that it is able to scale smoothly between the classical ballistic and diffusive extremes. Finally, we show that the probability distribution of the number of unique encounters in some time interval can be closely approximated by a Poisson distribution whose mean is given by our encounter model.
