# American Institute of Mathematical Sciences

August  2015, 20(6): 1639-1662. doi: 10.3934/dcdsb.2015.20.1639

## Time-invariant and stochastic disperser-structured matrix models: Invasion rates of fleshy-fruited exotic shrubs

 1 University of Miami, Institute of Theoretical and Mathematical Ecology, Department of Biology, P.O. Box 249118, Coral Gables, FL 33124-0421, United States, United States 2 United States Department of Agriculture, Plant Protection and Quarantine, Plant Epidemiology and Risk Analysis Laboratory, 1730 Varsity Drive, Suite 300, Raleigh, NC 27606-5202, United States

Received  November 2013 Revised  December 2014 Published  June 2015

Interest in spatial population dynamics includes applications to the spread of disease and invasive species. Recently, models for structured populations have been extended to incorporate temporal variation in both demography and dispersal. Here we propose a novel version of the model that incorporates structured dispersal to evaluate how changes in the relative proportion of mammalian, and short- and long-distance avian dispersers affect the rate of spread of an invasive shrub, Ardisia elliptica in Everglades National Park. We implemented $45$ time-invariant models, including one in which a single dispersal kernel was estimated from field data by pooling all seedlings, and $44$ that were disperser-structured in which dispersal kernels were estimated separately for gravity-, catbird-, robin- and raccoon-dispersed seed. Robins, the longest distance dispersers, are infrequent. Finally we implemented a time-varying model that included variability among years in the proportion of seeds that were taken by robins. The models estimated invasion speeds that ranged from $3.9$ to $34.7$ m $yr^{-1}$ . Infrequent long-distance dispersal by robins were important in determining invasion speed in the disperser-structured model. Comparing model projections with the (historically) known rate of spread, we show how a model that stratifies seeds by dispersal agents does better than one that ignores them, although all of our models underestimate it.
Citation: Carol C. Horvitz, Anthony L. Koop, Kelley D. Erickson. Time-invariant and stochastic disperser-structured matrix models: Invasion rates of fleshy-fruited exotic shrubs. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1639-1662. doi: 10.3934/dcdsb.2015.20.1639
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