Bounding Seed Loss from Isolated Habitat Patches.

Dispersal of propagules (seeds, spores) from a geographically isolated habitat into an uninhabitable matrix can play a decisive role in driving population dynamics. ODE and integrodifference models of these dynamics commonly feature a "dispersal success" parameter representing the average proportion of dispersing propagules that remain in viable habitat. While dispersal success can be estimated by empirical measurements or by integration of dispersal kernels, one may lack resources for fieldwork or details on dispersal kernels for numerical computation. Here we derive simple upper bounds on the proportion of propagule loss-the complement of dispersal success-that require only habitat area, habitat perimeter, and the mean dispersal distance of a propagule. Using vector calculus in a probabilistic framework, we rigorously prove bounds for the cases of both symmetric and asymmetric dispersal. We compare the bounds to simulations of integral models for the population of Asclepias syriaca (common milkweed) at McKnight Prairie-a 14 hectare reserve surrounded by agricultural fields in Goodhue County, Minnesota-and identify conditions under which the bounds closely estimate propagule loss.