Towards a general formalization of encounter rates in ecology

Although encounters between organisms are fundamental to many ecological processes, a general theory of encounters that accounts for random movements and probabilistic events has yet to be proposed. We present a framework for examining probabilistic encounters between arbitrarily moving searchers and immobile targets in continuous space and time. We define and contrast first encounter rates and mean encounter rates, which are generally not equal and depend on several properties of the process, including movement behaviors, the spatial scales of the encounter kernel, spatial distribution and birth-death dynamics of targets, and whether the encounters are destructive. Based on these considerations, we propose a taxonomy of encounter processes and discuss their functional relationships. Analytical approximations in several special cases are derived, leading to inference about general patterns. We identify, for example, cases (nondestructive, mean encounters) in which encounter rates are completely independent of movement velocity or tortuosity, and we quantify the dependence for cases (e.g., hard, first encounters and destructive encounters in a dynamic landscape) where there is a relationship. The analytical results lead to general qualitative conclusions, while the mathematical formalization and taxonomic organization provides a framework for studying and contrasting a broad range of encounter processes in ecology.