EMPIRICAL PATTERNS OF GEOGRAPHIC RANGE SIZE EXPANSION AND CONTRACTION IN MARINE INVERTEBRATES AND TERRESTRIAL MAMMALS ARE WELL PREDICTED BY A RANDOM WALK

Understanding the determinants of geographic range size is a fundamental goal in biogeography. Nevertheless, despite a large body of research on the subject, most scientific findings are case-specific and few general rules have emerged. Here we use data on the geographic range sizes of extant marine invertebrate and terrestrial mammal genera, in combination with fossil occurrence data, to show that many features in both present-day and fossil-derived geographic range sizes are well predicted by a Gaussian random walk. This model makes only two assumptions: 1) there is an absorbing boundary at a range size of zero representing origination and extinction, and 2) there is an equal probability of expansion and contraction in range size in each time-increment. The models are calibrated on the strong linear relationships that are observed between cohort ages and variance in geographic range sizes, regardless of whether fossil or extant ranges are used. Empirically-calibrated random walk models predict three commonly discussed geographic range size phenomena: 1) lineages with large geographic ranges are more likely to be older than lineages with narrow geographic ranges, 2) lineages on average first expand and then contract in geographic range over their lifespans, and 3) geographic range size distributions are approximately lognormal. We find a strong match between model predictions and empirical data for both marine invertebrates and mammals, though mammals have a much steeper variance-age linear slope, implying greater intrinsic rates of geographic range shift. We interpret model-data agreement as evidence that the responses of lineages to continually evolving physical and biological boundary conditions are sufficiently complex and individualistic that, when combined, general properties of the central limit theorem apply. Random walks are therefore a promising null model for future studies of geographic range size evolution.