Investigating first-crossing statistics in movement models with home-ranging behavior

Ecological populations are, in general, not well mixed, and their non-homogeneous use of space modulates their local interactions. Although this range-residency is known to affect important observables such as encounter rates between individuals, many models in population dynamics implicitly or explicitly assume that populations make homogenous use of space. The Ornstein-Uhlenbeck (OU) process is a stochastic process in space that displays the basic characteristics of movement bounded by and centered around a home-range. Within the framework of OU movement models, the crossing statistics of two simultaneous processes serve as a proxy for the encounter statistics of two individuals. While this mathematical problem has been investigated, especially in one dimension, fewer studies have addressed the more ecologically relevant two-dimensional case (2D). In this work, we conduct a numerical and semi-analytical study of the first-crossing statistics of a pair of 2D OU models. We believe this can help build the foundations of more mechanistic and realistic models of population dynamics, based on the scaling properties of individual interactions. Further analytical investigation of this problem may elucidate its general properties.