American Institute of Mathematical Sciences

Advanced
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 and 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), 407-421. doi: 10.1007/s12080-010-0091-z.

[2]

D. A. Cimprich and F. R. Moore, Gray Catbird, The Birds of North America, 5 (1995), 1-20.

[3]

J. S. Clark, E. Macklin and L. Wood, Stages and spatial scales of recruitment limitation in southern Appalachian forests, Ecological Monographs, 68 (1998), 213-235.

[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), 1475-1494. doi: 10.2307/176541.

[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), 283-298. doi: 10.1016/j.tpb.2012.03.005.

[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, Everglades National Park, South Florida Research Center, 1982.

[7]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[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, D. C. Schmitz and T. C. Brown), Island Press, 1997, 21-37.

[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), 464-475. doi: 10.1086/303193.

[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), 194-198. doi: 10.2307/3802990.

[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), 86-97. doi: 10.1890/09-2226.1.

[12]

A. L. Koop, Population Dynamics and Invasion Rate of an Invasive, Tropical Understory Shrub, Ardisia Elliptica, Ph.D dissertation, University of Miami, 2003.

[13]

A. L. Koop, Differential seed mortality among habitats limits the distribution of the invasive non-native shrub Ardisia elliptica, Plant Ecology, 172 (2004), 237-249.

[14]

A. L. Koop and C. C. Horvitz, Projection matrix analysis of the demography of an invasive, nonnative shrub (Ardisia elliptica), Ecology, 86 (2005), 2661-2672.

[15]

M. Kot, Discrete-time travelling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436. doi: 10.1007/BF00173295.

[16]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. doi: 10.2307/2265698.

[17]

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

[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), G711-G718.

[19]

D. J. Levey, J. J. Tewksbury and B. M. Bolker, Modelling long-distance seed dispersal in heterogeneous landscapes, Journal of Ecology, 96 (2008), 599-608. doi: 10.1111/j.1365-2745.2008.01401.x.

[20]

W. M. Lonsdale, Rates of spread of an invading species - Mimosa pigra in northern Australia, Journal of Ecology, 81 (1993), 513-521.

[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), 173-186. doi: 10.2307/1368446.

[22]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.

[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), 2nd edition, Springer, 1998, 283-305. doi: 10.1007/978-1-4615-6051-7_11.

[24]

J. H. Rappole and D. W. Warner, Ecological aspects of migrant bird behavior in Veracruz, Mexico, in Migrant Birds in the Neotropics: Ecology, Behavior, Distribution and Conservation (eds. A. Keast and E. S. Morton), Smithsonian Institution, Washington D. C., 1980, 343-394.

[25]

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

[26]

K. Shea and D. Kelly, Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand, Ecological Applications, 8 (1998), 824-832.

[27]

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

[28]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[29]

M. B. Soons and J. Bullock, Non-random seed abscission, long-distance wind dispersal and plant migration rates, Journal of Ecology, 96 (2008), 581-590. doi: 10.1111/j.1365-2745.2008.01370.x.

[30]

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

[31]

R. P. Wunderlin, Guide to the Vascular Plants of Florida, University Press of Florida, Gainesville, Florida, 1998.

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), 407-421. doi: 10.1007/s12080-010-0091-z.

[2]

D. A. Cimprich and F. R. Moore, Gray Catbird, The Birds of North America, 5 (1995), 1-20.

[3]

J. S. Clark, E. Macklin and L. Wood, Stages and spatial scales of recruitment limitation in southern Appalachian forests, Ecological Monographs, 68 (1998), 213-235.

[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), 1475-1494. doi: 10.2307/176541.

[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), 283-298. doi: 10.1016/j.tpb.2012.03.005.

[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, Everglades National Park, South Florida Research Center, 1982.

[7]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[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, D. C. Schmitz and T. C. Brown), Island Press, 1997, 21-37.

[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), 464-475. doi: 10.1086/303193.

[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), 194-198. doi: 10.2307/3802990.

[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), 86-97. doi: 10.1890/09-2226.1.

[12]

A. L. Koop, Population Dynamics and Invasion Rate of an Invasive, Tropical Understory Shrub, Ardisia Elliptica, Ph.D dissertation, University of Miami, 2003.

[13]

A. L. Koop, Differential seed mortality among habitats limits the distribution of the invasive non-native shrub Ardisia elliptica, Plant Ecology, 172 (2004), 237-249.

[14]

A. L. Koop and C. C. Horvitz, Projection matrix analysis of the demography of an invasive, nonnative shrub (Ardisia elliptica), Ecology, 86 (2005), 2661-2672.

[15]

M. Kot, Discrete-time travelling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436. doi: 10.1007/BF00173295.

[16]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. doi: 10.2307/2265698.

[17]

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

[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), G711-G718.

[19]

D. J. Levey, J. J. Tewksbury and B. M. Bolker, Modelling long-distance seed dispersal in heterogeneous landscapes, Journal of Ecology, 96 (2008), 599-608. doi: 10.1111/j.1365-2745.2008.01401.x.

[20]

W. M. Lonsdale, Rates of spread of an invading species - Mimosa pigra in northern Australia, Journal of Ecology, 81 (1993), 513-521.

[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), 173-186. doi: 10.2307/1368446.

[22]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.

[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), 2nd edition, Springer, 1998, 283-305. doi: 10.1007/978-1-4615-6051-7_11.

[24]

J. H. Rappole and D. W. Warner, Ecological aspects of migrant bird behavior in Veracruz, Mexico, in Migrant Birds in the Neotropics: Ecology, Behavior, Distribution and Conservation (eds. A. Keast and E. S. Morton), Smithsonian Institution, Washington D. C., 1980, 343-394.

[25]

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

[26]

K. Shea and D. Kelly, Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand, Ecological Applications, 8 (1998), 824-832.

[27]

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

[28]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[29]

M. B. Soons and J. Bullock, Non-random seed abscission, long-distance wind dispersal and plant migration rates, Journal of Ecology, 96 (2008), 581-590. doi: 10.1111/j.1365-2745.2008.01370.x.

[30]

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

[31]

R. P. Wunderlin, Guide to the Vascular Plants of Florida, University Press of Florida, Gainesville, Florida, 1998.

[1]

Mohammad El Smaily, François Hamel, Lionel Roques. Homogenization and influence of fragmentation in a biological invasion model. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 321-342. doi: 10.3934/dcds.2009.25.321
[2]

Peng Zhong, Suzanne Lenhart. Optimal control of integrodifference equations with growth-harvesting-dispersal order. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2281-2298. doi: 10.3934/dcdsb.2012.17.2281
[3]

Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
[4]

Franziska Hinkelmann, Reinhard Laubenbacher. Boolean models of bistable biological systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1443-1456. doi: 10.3934/dcdss.2011.4.1443
[5]

Jon Jacobsen, Taylor McAdam. A boundary value problem for integrodifference population models with cyclic kernels. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3191-3207. doi: 10.3934/dcdsb.2014.19.3191
[6]

Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2771-2788. doi: 10.3934/dcdsb.2012.17.2771
[7]

Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221
[8]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761
[9]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
[10]

Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146
[11]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[12]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032
[13]

Risei Kano. The existence of solutions for tumor invasion models with time and space dependent diffusion. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 63-74. doi: 10.3934/dcdss.2014.7.63
[14]

Risei Kano, Akio Ito. The existence of time global solutions for tumor invasion models with constraints. Conference Publications, 2011, 2011 (Special) : 774-783. doi: 10.3934/proc.2011.2011.774
[15]

Jian-Wen Sun, Wan-Tong Li, Zhi-Cheng Wang. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217
[16]

Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409
[17]

Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199
[18]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
[19]

Nalin Fonseka, Ratnasingham Shivaji, Jerome Goddard, Ⅱ, Quinn A. Morris, Byungjae Son. On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3401-3415. doi: 10.3934/dcdss.2020245
[20]

Mary P. Hebert, Linda J. S. Allen. Disease outbreaks in plant-vector-virus models with vector aggregation and dispersal. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2169-2191. doi: 10.3934/dcdsb.2016042

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy