Scale dependence of immigration rates: models, metrics and data

We examine the relationship between immigration rate and patch area for different types of movement behaviours and detection modes. Theoretical models suggest that the scale dependence of the immigration rate per unit area (I/A) can be described by a power model I/A = i*Area(zeta), where zeta describes the strength of the scale dependence. Three types of scaling were identified. Area scaling (zeta = 0) is expected for passively dispersed organisms that have the same probability of landing anywhere in the patch. Perimeter scaling (-0.30 > zeta > -0.45) is expected when patches are detected from a very short distance and immigrants arrive over the patch boundary, whereas diameter scaling (zeta = -0.5) is expected if patches are detected from a long distance or if search is approximately linear. A meta-analysis of published empirical studies of the scale dependence of immigration rates in terrestrial insects suggests that butterflies show diameter scaling, aphids show area scaling, and the scaling of beetle immigration is highly variable. We conclude that the scaling of immigration rates in many cases can be predicted from search behaviour and the mode of patch detection.