AM  Vol.6 No.9 , August 2015
Mathematical Model of Seed Dispersal by Frugivorous Birds and Migration Potential of Pinyon and Juniper in Utah
Abstract: Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transported and settled seeds can be described by a dispersal kernel, which captures the probability that the seed will move a certain distance by the end of the process. To model active seed dispersal of this nature, we introduce handling time probabilities into the dispersal model to generate a seed digestion kernel. In the limit of no variability in handling time the seed digestion kernel is Gaussian, whereas for uniform variability in handling time the kernel approaches a Laplace distribution. This allows us to standardize spatial movement (diffusion) and handling time (peak settling rate) parameters for all three distributions and compare. Analysis of the tails indicates that the seed digestion kernel decays at a rate intermediate between Gaussian and Laplace seed kernels. Using this seed digestion kernel, we create an invasion model to estimate the speed at which juniper and pinyon forest boundaries move. We find that the speed of seed invasion corresponding to the digestion kernel was faster than seeds resulting from Laplace and Gaussian kernels for more rapidly digested seeds. For longer handling times the speeds are bounded between the Laplace (faster) and Gaussian (slower) speeds. Using parameter values from the literature we evaluate the migration potential of pinyon and juniper, finding that pinyon may be able to migrate up to two orders of magnitude more rapidly, consistent with observations of pine migration during the Holocene.
Cite this paper: Neupane, R. and Powell, J. (2015) Mathematical Model of Seed Dispersal by Frugivorous Birds and Migration Potential of Pinyon and Juniper in Utah. Applied Mathematics, 6, 1506-1523. doi: 10.4236/am.2015.69135.

[1]   Corlett, R.T. (1998) Furgivory and Seed Dispersal by Vertebrates in the Oriental (Indomalayan) Region. Cambridge Philosophical Society, 73, 413-448.

[2]   Wenny, D.G. (2001) Advantage of Seed Dispersal: A Re-Evaluation of Directed Dispersal. Evolutionary Ecology Research, 3, 51-74.

[3]   Clark, C.J., Poulsen, J.R. and Parker, V.T. (2001) The Role of Arboreal Seed Dispersal Groups on the Seed Rain of a Lowland Tropical Forest. Biotropica, 33, 606-620.

[4]   Herrera, C.M. (1995) Plant-Vertebrate Seed Dispersal Systems in the Mediterranean: Ecological Evolutionary and Historical Determinants. Annual Review of Ecology and Systematics, 26, 705-727.

[5]   Chambers, J.C., Vander Wall, S.B. and Schupp, E.W. (1999) Seed and Seedling Ecology of Pinyon and Juniper Species in the Pygmy Woodland of Western North America. Botanical Review, 65, 1-38.

[6]   Chavez-Ramirez, F. and Slack, R.D. (1994) Effects of Avian Foraging and Post-Foraging Behavior on Seed Dispersal Patterns of Ashe Juniper. Nordic Society Oikos, 71, 40-46.

[7]   Vander Wall, S.B. and Balda, R.P. (1977) Coadaptations of the Clark’s Nutcracker and the Pinon Pine for Efficient Seed Harvest and Dispersal. Ecology, 47, 89-111.

[8]   Balda, R.P. and Bateman, G.C. (1971) Flocking and Annual Cycle of Pinon Jay, Gymnorhinus cyanocephalus. Cooper Ornithological Society, 73, 287-302.

[9]   Neilson, K.P. (1987) On the Interface between Current Ecological Studies and the Paleobotany of Pinyon-Juniper Woodlands. In: Everett, R.L., Ed., Proceeding of the Pinyon-Juniper Conference, General Technical Report INT-215, US Department of Agriculture, Forest Service, Intermountain Research Station, Reno, 93-98.

[10]   Miller, R.F. and Wigand, P.E. (1994) Holocene Changes in Semiarid Pinyon-Juniper Woodlands. BioScience, 44, 465-474.

[11]   Breshears, D.D., Cobb, N.S., Rich, P.M., Price, K.P., Allen, C.D., Balice, R.G., Romme, W.H., Kastens, J.H., Floyd, L.M., Belnap, J., Anderson, J.J., Myers, O.B. and Meyer, C.W. (2005) Regional Vegetation Die-Off in Response to Global-Change-Type Drought. Proceedings of the National Academy of Sciences of the United States of America, 102, 15144-15148.

[12]   Weisberg, P.J., Lingua, E. and Pillai, R.B. (2007) Spatial Patterns of Pinyon-Juniper Woodland Expansion in Central Nevada. Rangeland Ecology & Management, 60, 115-124.

[13]   Mueller, R.C., Scudder, C.M., Porter, M.E., Trotter III, T.R., Gehring, C.A. and Whitham, T.G. (2005) Differential Tree Mortality in Response to Severe Drought: Evidence for Long-Term Vegetation Shifts. Journal of Ecology, 93, 1085-1093.

[14]   Breshears, D.D., Myers, O.B., Johnson, S.R., Meyer, C.W. and Martens, S.N. (1997) Differential Use of Spatially Heterogeneous Soil Moisture by Two Semiarid Woody Species: Pinus edulis and Juniperus monosperma. Journal of Ecology, 85, 289-299.

[15]   Allen, C.D. and Breshears, D.D. (1998) Drought-Induced Shift of a Forest-Woodland Ecotone: Rapid Landscape Response to Climate Variation. Proceedings of the National Academy of Sciences of the United States of America, 95, 14839-14842.

[16]   Bradley, B.A. and Fleishman, E. (2008) Relationships between Expanding Pinyon-Juniper Cover and Topography in the Central Great Basin, Nevada. Journal of Biogeography, 35, 951-964.

[17]   Gray, S.T., Betancourt, J.L., Jackson, S.T. and Eddy, R.G. (2006) Role of Multidecadal Climate Variability in a Range Extension of Pinyon Pine. Ecology, 87, 1124-1130.[1124:ROMCVI]2.0.CO;2

[18]   Miller, R.E. and Rose, J.A. (1995) Historic Expansion of Juniperus occidentalis (Western Juniper) in Southeastern Oregon. Great Basin Naturalist, 55, 37-45.

[19]   Neubert, M.G., Kot, M. and Lewis, M.A. (1995) Dispersal and Pattern Formation in a Discrete-Time Predator-Prey Model. Theoretical Population Biology, 48, 7-43.

[20]   Holthuijzen, A.M.A. and Adkisson, C.S. (1984) Passage Rate, Energetics, and Utilization Efficiency of the Cedar Waxwing. The Wilson Bulletin, 96, 680-684.

[21]   Marsden, J.E. and Hoffman, M.J. (1987) Basic Complex Analysis. W.H. Freeman and Company, New York.

[22]   Powell, J.A. and Zimmermann, N.E. (2004) Multiscale Analysis of Active Seed Dispersal Contributes to Resolving Reid’s Paradox. Ecology, 85, 490-506.

[23]   Kot, M., Lewis, M.A. and van den Driessche, P. (1996) Dispersal Data and Spread of Invading Organisms. Ecology, 77, 2027-2042.

[24]   Tausch, R.J. and West, N.E. (1988) Differential Establishment of Pinyon and Juniper Following Fire. American Midland Naturalist, 119, 174-184.

[25]   Turchin, P. (1998) Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution of Plants and Animals. Sinauer Associates, Sunderland, MA.

[26]   Suzan-Azpiri, H., Sanchez-Ramos, G., Martinez-Avalos, J.G., Villa-Melgarejo, S. and Franco, M. (2002) Population Structure of Pinus nelsoni Shaw, an Endemic Pinyon Pine in Tamaulipas, Mexico. Forest Ecology of Management, 165, 193-203.

[27]   Shaw, J.D., Steed, B.E. and DeBlander, L.T. (2005) Forest Inventory and Analysis (FIA) Annual Inventory Answers the Question: What Is Happening to Pinyon-Juniper Woodlands? Journal of Forestry, 103, 280-285.

[28]   Li, Z., Zou, J., Mao, K., Lin, K., Li, H., Liu, J., Kallman, T. and Lascoux, M. (2011) Population Genetic Evidence for Complex Evolutionary Histories of Four High Altitude Juniper Species in the Qinghai-Tibetan Plateau. Evolution, 66, 831-845.

[29]   Nowak, C.L., Nowak, R.S., Tausch, R.J. and Wigand, P.E. (1994) Tree and Shrub Dynamics in Northwestern Great Basin Woodland and Shrub Steppe during the Late-Pleistocene and Holocene. American Journal of Botany, 81, 265-277.

[30]   Lanner, R.M. and Van Devender, T.R. (1998) The Recent History of Pinyon Pines in the American Southwest. In: Richardson, D.M., Ed., Ecology and Biogeography of Pinus, The Press Syndicate of the University of Cambridge, Cambridge, 171-182.

[31]   Vander Wall, S.B. and Balda, R.P. (1997) Dispersal of Singleleaf Pinon Pine (Pinus monophylla) by Seed-Caching Rodents. Journal of Mammalogy, 78, 181-191.

[32]   Willson, M.F. (1993) Mammals as Seed-Dispersal Mutualists in North America. Oikos, 67, 159-176.

[33]   Garlick, M.J., Powell, J.A., Hooten, M.B. and McFarlane, L.R. (2010) Homogenization of Large-Scale Movement Models in Ecology. Society of Mathematical Biology, 73, 2088-2108.

[34]   Ascher, U.M. and Greif, C. (2011) A First Course in Numerical Methods. SIAM, Philadelphia.