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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

Carol C. Horvitz 1, , Anthony L. Koop 2, and  Kelley D. Erickson 1,

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.
Keywords: Everglades National Park, dispersal kernel, Florida., biological invasions, invasion rate, matrix models, integrodifference equation, Ardisia elliptica.
Mathematics Subject Classification: 91B72, 91B70, 91D25, 91D20, 92D25, 92D40, 60H30, 60J10, 60J2.
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
References:
[1]

H. Caswell, M. G. Neubert and C. M. Hunter, Demography and dispersal: Invasion speeds and sensitivity analysis in periodic and stochastic environments,, Theoretical Ecology, 4 (2011), 407.  doi: 10.1007/s12080-010-0091-z.  Google Scholar

[2]

D. A. Cimprich and F. R. Moore, Gray Catbird,, The Birds of North America, 5 (1995), 1.   Google Scholar

[3]

J. S. Clark, E. Macklin and L. Wood, Stages and spatial scales of recruitment limitation in southern Appalachian forests,, Ecological Monographs, 68 (1998), 213.   Google Scholar

[4]

J. S. Clark, M. Silman, R. Kern, E. Macklin and J. HilleRisLambers, Seed dispersal near and far: Patterns across temperate and tropical forests,, Ecology, 80 (1999), 1475.  doi: 10.2307/176541.  Google Scholar

[5]

S. P. Ellner and S. J. Schreiber, Temporally variable dispersal and demography can accelerate the spread of invading species,, Theoretical Population Biology, 82 (2012), 283.  doi: 10.1016/j.tpb.2012.03.005.  Google Scholar

[6]

J. J. Ewel, D. S. Ojima, D. A. Karl and W. F. DeBusk, Schinus in Successional Ecosystems of Everglades National Park,, Technical Report T-676, (1982).   Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[8]

D. R. Gordon and K. P. Thomas, Florida's invasion by nonindigenous plants: history, screening, and regulation,, in Strangers in Paradise (eds. D. Simberloff, (1997), 21.   Google Scholar

[9]

S. I. Higgins and D. M. Richardson, Predicting plant migration rates in a changing world: The role of long-distance dispersal,, American Naturalist, 153 (1999), 464.  doi: 10.1086/303193.  Google Scholar

[10]

K. M. Hodges, J. Chamberlain and B. D. Leopold, Effects of summer hunting on ranging behavior of adult raccoons in central Mississippi,, Journal of Wildlife Management, 64 (2000), 194.  doi: 10.2307/3802990.  Google Scholar

[11]

E. Jongejans, K. Shea, O. Skarpaas, D. Kelly and S. P. Ellner, Importance of individual and environmental variation for invasive species spread: A spatial integral projection model,, Ecology, 92 (2011), 86.  doi: 10.1890/09-2226.1.  Google Scholar

[12]

A. L. Koop, Population Dynamics and Invasion Rate of an Invasive, Tropical Understory Shrub, Ardisia Elliptica,, Ph.D dissertation, (2003).   Google Scholar

[13]

A. L. Koop, Differential seed mortality among habitats limits the distribution of the invasive non-native shrub Ardisia elliptica,, Plant Ecology, 172 (2004), 237.   Google Scholar

[14]

A. L. Koop and C. C. Horvitz, Projection matrix analysis of the demography of an invasive, nonnative shrub (Ardisia elliptica),, Ecology, 86 (2005), 2661.   Google Scholar

[15]

M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[16]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.  doi: 10.2307/2265698.  Google Scholar

[17]

K. A. Langeland and K. C. Burks, Identification and Biology of Non-Native Plants in Florida's Natural Areas,, University of Florida, (1998).   Google Scholar

[18]

D. J. Levey and W. H. Karasov, Digestive modulation in a seasonal frugivore, the American robin (Turdus migratorius),, American Journal of Physiology, 262 (1992).   Google Scholar

[19]

D. J. Levey, J. J. Tewksbury and B. M. Bolker, Modelling long-distance seed dispersal in heterogeneous landscapes,, Journal of Ecology, 96 (2008), 599.  doi: 10.1111/j.1365-2745.2008.01401.x.  Google Scholar

[20]

W. M. Lonsdale, Rates of spread of an invading species - Mimosa pigra in northern Australia,, Journal of Ecology, 81 (1993), 513.   Google Scholar

[21]

P. K. Malmborg and M. F. Willson, Foraging ecology of avian frugivores and some consequences for seed dispersal in an Illinois woodlot,, The Condor, 90 (1988), 173.  doi: 10.2307/1368446.  Google Scholar

[22]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations,, Ecology, 81 (2000), 1613.   Google Scholar

[23]

I. M. Parker and S. H. Reichard, Critical issues in invasion biology for conservation science,, in Conservation Biology for the Coming Decade (eds. P. L. Fiedler and P. M. Kareiva), (1998), 283.  doi: 10.1007/978-1-4615-6051-7_11.  Google Scholar

[24]

J. H. Rappole and D. W. Warner, Ecological aspects of migrant bird behavior in Veracruz, Mexico,, in Migrant Birds in the Neotropics: Ecology, (1980), 343.   Google Scholar

[25]

R. Seavey and J. Seavey, Ardisia Elliptica in Everglades National Park: An Overview Through 1993,, Unpublished Manuscript., ().   Google Scholar

[26]

K. Shea and D. Kelly, Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand,, Ecological Applications, 8 (1998), 824.   Google Scholar

[27]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford University Press, (1997).   Google Scholar

[28]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[29]

M. B. Soons and J. Bullock, Non-random seed abscission, long-distance wind dispersal and plant migration rates,, Journal of Ecology, 96 (2008), 581.  doi: 10.1111/j.1365-2745.2008.01370.x.  Google Scholar

[30]

S. Tuljapurkar, Population Dynamics in Variable Environments,, Lecture Notes in Biomathematics, (1990).  doi: 10.1007/978-3-642-51652-8.  Google Scholar

[31]

R. P. Wunderlin, Guide to the Vascular Plants of Florida,, University Press of Florida, (1998).   Google Scholar

show all references

References:
[1]

H. Caswell, M. G. Neubert and C. M. Hunter, Demography and dispersal: Invasion speeds and sensitivity analysis in periodic and stochastic environments,, Theoretical Ecology, 4 (2011), 407.  doi: 10.1007/s12080-010-0091-z.  Google Scholar

[2]

D. A. Cimprich and F. R. Moore, Gray Catbird,, The Birds of North America, 5 (1995), 1.   Google Scholar

[3]

J. S. Clark, E. Macklin and L. Wood, Stages and spatial scales of recruitment limitation in southern Appalachian forests,, Ecological Monographs, 68 (1998), 213.   Google Scholar

[4]

J. S. Clark, M. Silman, R. Kern, E. Macklin and J. HilleRisLambers, Seed dispersal near and far: Patterns across temperate and tropical forests,, Ecology, 80 (1999), 1475.  doi: 10.2307/176541.  Google Scholar

[5]

S. P. Ellner and S. J. Schreiber, Temporally variable dispersal and demography can accelerate the spread of invading species,, Theoretical Population Biology, 82 (2012), 283.  doi: 10.1016/j.tpb.2012.03.005.  Google Scholar

[6]

J. J. Ewel, D. S. Ojima, D. A. Karl and W. F. DeBusk, Schinus in Successional Ecosystems of Everglades National Park,, Technical Report T-676, (1982).   Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[8]

D. R. Gordon and K. P. Thomas, Florida's invasion by nonindigenous plants: history, screening, and regulation,, in Strangers in Paradise (eds. D. Simberloff, (1997), 21.   Google Scholar

[9]

S. I. Higgins and D. M. Richardson, Predicting plant migration rates in a changing world: The role of long-distance dispersal,, American Naturalist, 153 (1999), 464.  doi: 10.1086/303193.  Google Scholar

[10]

K. M. Hodges, J. Chamberlain and B. D. Leopold, Effects of summer hunting on ranging behavior of adult raccoons in central Mississippi,, Journal of Wildlife Management, 64 (2000), 194.  doi: 10.2307/3802990.  Google Scholar

[11]

E. Jongejans, K. Shea, O. Skarpaas, D. Kelly and S. P. Ellner, Importance of individual and environmental variation for invasive species spread: A spatial integral projection model,, Ecology, 92 (2011), 86.  doi: 10.1890/09-2226.1.  Google Scholar

[12]

A. L. Koop, Population Dynamics and Invasion Rate of an Invasive, Tropical Understory Shrub, Ardisia Elliptica,, Ph.D dissertation, (2003).   Google Scholar

[13]

A. L. Koop, Differential seed mortality among habitats limits the distribution of the invasive non-native shrub Ardisia elliptica,, Plant Ecology, 172 (2004), 237.   Google Scholar

[14]

A. L. Koop and C. C. Horvitz, Projection matrix analysis of the demography of an invasive, nonnative shrub (Ardisia elliptica),, Ecology, 86 (2005), 2661.   Google Scholar

[15]

M. Kot, Discrete-time travelling waves: Ecological examples,, Journal of Mathematical Biology, 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[16]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.  doi: 10.2307/2265698.  Google Scholar

[17]

K. A. Langeland and K. C. Burks, Identification and Biology of Non-Native Plants in Florida's Natural Areas,, University of Florida, (1998).   Google Scholar

[18]

D. J. Levey and W. H. Karasov, Digestive modulation in a seasonal frugivore, the American robin (Turdus migratorius),, American Journal of Physiology, 262 (1992).   Google Scholar

[19]

D. J. Levey, J. J. Tewksbury and B. M. Bolker, Modelling long-distance seed dispersal in heterogeneous landscapes,, Journal of Ecology, 96 (2008), 599.  doi: 10.1111/j.1365-2745.2008.01401.x.  Google Scholar

[20]

W. M. Lonsdale, Rates of spread of an invading species - Mimosa pigra in northern Australia,, Journal of Ecology, 81 (1993), 513.   Google Scholar

[21]

P. K. Malmborg and M. F. Willson, Foraging ecology of avian frugivores and some consequences for seed dispersal in an Illinois woodlot,, The Condor, 90 (1988), 173.  doi: 10.2307/1368446.  Google Scholar

[22]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations,, Ecology, 81 (2000), 1613.   Google Scholar

[23]

I. M. Parker and S. H. Reichard, Critical issues in invasion biology for conservation science,, in Conservation Biology for the Coming Decade (eds. P. L. Fiedler and P. M. Kareiva), (1998), 283.  doi: 10.1007/978-1-4615-6051-7_11.  Google Scholar

[24]

J. H. Rappole and D. W. Warner, Ecological aspects of migrant bird behavior in Veracruz, Mexico,, in Migrant Birds in the Neotropics: Ecology, (1980), 343.   Google Scholar

[25]

R. Seavey and J. Seavey, Ardisia Elliptica in Everglades National Park: An Overview Through 1993,, Unpublished Manuscript., ().   Google Scholar

[26]

K. Shea and D. Kelly, Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand,, Ecological Applications, 8 (1998), 824.   Google Scholar

[27]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford University Press, (1997).   Google Scholar

[28]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[29]

M. B. Soons and J. Bullock, Non-random seed abscission, long-distance wind dispersal and plant migration rates,, Journal of Ecology, 96 (2008), 581.  doi: 10.1111/j.1365-2745.2008.01370.x.  Google Scholar

[30]

S. Tuljapurkar, Population Dynamics in Variable Environments,, Lecture Notes in Biomathematics, (1990).  doi: 10.1007/978-3-642-51652-8.  Google Scholar

[31]

R. P. Wunderlin, Guide to the Vascular Plants of Florida,, University Press of Florida, (1998).   Google Scholar

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