January  2011, 10(1): 339-359. doi: 10.3934/cpaa.2011.10.339

On the global attractor of the Trojan Y Chromosome model

1. 

Clarkson University, Department of Mathematics & Computer Science, 106 Technology Advancement Center, Potsdam, NY 13676, USA Government

2. 

Mathematical Biosciences Institute, The Ohio State University, 1735 Neil Ave, 3rd Floor, Columbus, OH 43210, USA Government

Received  January 2010 Revised  June 2010 Published  November 2010

We consider the Trojan Y Chromosome (TYC) model for eradication of invasive species in population dynamics. We present global estimates for the TYC system in a spatial domain. In this work we prove the existence of a global attractor for the system. We derive uniform estimates to tackle the question of asymptotic compactness of the semi-group for the TYC model in $H^2(\Omega)$. This along with the existence of a bounded absorbing set, which we also derive, demonstrates the existence of a global attractor for the TYC model. The present analysis reveals that extinction of an invasive species is always possible to achieve irrespective of geometric considerations of the domain. This result is valid for TYC systems in which advection is negligible. This theoretical work lays the foundation for experimental studies of the application of the TYC eradication strategy in spatial ecology, since the outcome is in principle guaranteed.
Citation: Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure and Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339
References:
[1]

P. Brown and T. I. Walker, CARPSIM: stochastic simulation modelling of wild carp (Cyprinus carpio L.) population dynamics, with applications to pest control, Ecological Modelling, 176 (2004), 83-97. doi: doi:10.1016/j.ecolmodel.2003.11.009.

[2]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley, West Sussex, England, 2003.

[3]

"Economic Research Service, United States Department of Agriculture,", Agricultural Outlook: Statistical Indicators, May 2009. Retrieved on 06/01/2009 from http://www.ers.usda.gov/publications/agoutlook/aotables/.

[4]

J. B. Gutierrez, "Mathematical Analysis of The Use Of Trojan Sex Chromosomes as Means of Eradication of Invasive Species," PhD thesis, Florida State University, Tallahassee, FL, 2009.

[5]

J. B. Gutierrez, M. K. Hurdal, R. D. Parshad and J. L. Teem, Analysis of the trojan y chromosome model for eradication of invasive species in a dendritic riverine system, Submitted for publication, 2010.

[6]

J. B. Gutierrez and J. L. Teem, A model describing the effect of sex-reversed YY fish in an established wild population: the use of a Trojan Y chromosome to cause extinction of an introduced exotic species, Journal of Theoretical Biology, 241 (2006), 333-341. doi: doi:10.1016/j.jtbi.2005.11.032.

[7]

J. E. Hill and C. E. Cichra, Eradication of a reproducing population of Convict Cichlids, Cichlasoma nigrofasciatum (Cichlidae) in North-Central Florida, Florida Scientist, 68 (2005), 65-74.

[8]

J. J. Hoover, K. J. Killgore and A. F. Cofrancesco, Suckermouth catfishes: threats to aquatic ecosystems of the United States? Aquatic Nuisance Species Research, 4 (2004), 1-9.

[9]

R. D. Howard, J. A. DeWoody and W. M. Muir, Transgenic male mating advantage provides opportunity for Trojan gene effect in a fish, Proceedings of the National Academy of Sciences, 101 (2004), 2934-2938. doi: doi:10.1073/pnas.0306285101.

[10]

M. A. Hurley, P. Matthiesen and A. D. Pickering, A model for environmental sex reversal in fish, Journal of Theoretical Biology, 227 (2004), 159-165. doi: doi:10.1016/j.jtbi.2003.10.010.

[11]

W. M. Muir and R. D. Howard, Possible ecological risks of transgenic organism release when transgenes affect mating success: sexual selection and the Trojan gene hypothesis, Proceedings of the National Academy of Sciences, 96 (1999), 13853-13856. doi: doi:10.1073/pnas.96.24.13853.

[12]

J. H. Myers, D. Simberloff, A. M. Kuris and J. R. Carey, Eradication revisited: dealing with exotic species, Trends in Ecology & Evolution, 15 (2000), 316-320. doi: doi:10.1016/S0169-5347(00)01914-5.

[13]

J. J. Nagler, J. Bouma, G. H. Thorgaard and D. D. Dauble, High incidence of a male-specific genetic marker in phenotypic female Chinook salmon from the Columbia river, Environmental Health Perspectives, 109 (2001), 67-69. doi: doi:10.2307/3434923.

[14]

O. T. A., Harmful non-indigenous species in the United States, OTA-F-565 U.S. Congress, Office of Technology Assessment, Washington, DC, 1993.

[15]

P. Palace, R. E. Evans, K. Wautierand L. Vandenbyllardt, W. W. Vandersteen and K. Kidd, Induction of vitellogenin and histological effects in wild fathead minnows from a lake experimentally treated with the synthetic estrogen, ethynylestradiol, Water Quality Research Journal Canada, 37 (2002), 637-650.

[16]

R. D. Parshad and J. B. Gutierrez, On the well-posedness of the TYC system, Submitted for publication, 2010.

[17]

D. Pimentel, R. Zuniga and D. Morrison, Update on the environmental and economic costs associated with alien-invasive species in the united states, Ecological Economics, 52 (2005), 273-288.

[18]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge University Press, New York, NY, 2001.

[19]

P. L. Shafland, Exotic fish assessments: an alternative view, Reviews in Fisheries Science, 4 (1996), 123-132. doi: doi:10.1080/10641269609388582.

[20]

P. L. Shafland, Exotic fishes of Florida 1994, Reviews in Fisheries Science, 4 (1996), 101-122. doi: doi:10.1080/10641269609388581.

[21]

P. L. Shafland and K. L. Foote, A reproducing population of Serrasalmus humeralis Valenciennes in southern Florida, Florida Scientist, 42 (1979), 206-214.

[22]

W. L. Shelton, Broodstock development for monosex production of grass carp, Aquaculture, 57 (1986), 311-319. doi: doi:10.1016/0044-8486(86)90209-7.

[23]

R. Temam, "Navier Stokes Equations and Nonlinear Functional Analysis," Society for Industrial and Applied Mathematics, Philadelphia, PA, 1985.

[24]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, NY, 1998.

show all references

References:
[1]

P. Brown and T. I. Walker, CARPSIM: stochastic simulation modelling of wild carp (Cyprinus carpio L.) population dynamics, with applications to pest control, Ecological Modelling, 176 (2004), 83-97. doi: doi:10.1016/j.ecolmodel.2003.11.009.

[2]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley, West Sussex, England, 2003.

[3]

"Economic Research Service, United States Department of Agriculture,", Agricultural Outlook: Statistical Indicators, May 2009. Retrieved on 06/01/2009 from http://www.ers.usda.gov/publications/agoutlook/aotables/.

[4]

J. B. Gutierrez, "Mathematical Analysis of The Use Of Trojan Sex Chromosomes as Means of Eradication of Invasive Species," PhD thesis, Florida State University, Tallahassee, FL, 2009.

[5]

J. B. Gutierrez, M. K. Hurdal, R. D. Parshad and J. L. Teem, Analysis of the trojan y chromosome model for eradication of invasive species in a dendritic riverine system, Submitted for publication, 2010.

[6]

J. B. Gutierrez and J. L. Teem, A model describing the effect of sex-reversed YY fish in an established wild population: the use of a Trojan Y chromosome to cause extinction of an introduced exotic species, Journal of Theoretical Biology, 241 (2006), 333-341. doi: doi:10.1016/j.jtbi.2005.11.032.

[7]

J. E. Hill and C. E. Cichra, Eradication of a reproducing population of Convict Cichlids, Cichlasoma nigrofasciatum (Cichlidae) in North-Central Florida, Florida Scientist, 68 (2005), 65-74.

[8]

J. J. Hoover, K. J. Killgore and A. F. Cofrancesco, Suckermouth catfishes: threats to aquatic ecosystems of the United States? Aquatic Nuisance Species Research, 4 (2004), 1-9.

[9]

R. D. Howard, J. A. DeWoody and W. M. Muir, Transgenic male mating advantage provides opportunity for Trojan gene effect in a fish, Proceedings of the National Academy of Sciences, 101 (2004), 2934-2938. doi: doi:10.1073/pnas.0306285101.

[10]

M. A. Hurley, P. Matthiesen and A. D. Pickering, A model for environmental sex reversal in fish, Journal of Theoretical Biology, 227 (2004), 159-165. doi: doi:10.1016/j.jtbi.2003.10.010.

[11]

W. M. Muir and R. D. Howard, Possible ecological risks of transgenic organism release when transgenes affect mating success: sexual selection and the Trojan gene hypothesis, Proceedings of the National Academy of Sciences, 96 (1999), 13853-13856. doi: doi:10.1073/pnas.96.24.13853.

[12]

J. H. Myers, D. Simberloff, A. M. Kuris and J. R. Carey, Eradication revisited: dealing with exotic species, Trends in Ecology & Evolution, 15 (2000), 316-320. doi: doi:10.1016/S0169-5347(00)01914-5.

[13]

J. J. Nagler, J. Bouma, G. H. Thorgaard and D. D. Dauble, High incidence of a male-specific genetic marker in phenotypic female Chinook salmon from the Columbia river, Environmental Health Perspectives, 109 (2001), 67-69. doi: doi:10.2307/3434923.

[14]

O. T. A., Harmful non-indigenous species in the United States, OTA-F-565 U.S. Congress, Office of Technology Assessment, Washington, DC, 1993.

[15]

P. Palace, R. E. Evans, K. Wautierand L. Vandenbyllardt, W. W. Vandersteen and K. Kidd, Induction of vitellogenin and histological effects in wild fathead minnows from a lake experimentally treated with the synthetic estrogen, ethynylestradiol, Water Quality Research Journal Canada, 37 (2002), 637-650.

[16]

R. D. Parshad and J. B. Gutierrez, On the well-posedness of the TYC system, Submitted for publication, 2010.

[17]

D. Pimentel, R. Zuniga and D. Morrison, Update on the environmental and economic costs associated with alien-invasive species in the united states, Ecological Economics, 52 (2005), 273-288.

[18]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge University Press, New York, NY, 2001.

[19]

P. L. Shafland, Exotic fish assessments: an alternative view, Reviews in Fisheries Science, 4 (1996), 123-132. doi: doi:10.1080/10641269609388582.

[20]

P. L. Shafland, Exotic fishes of Florida 1994, Reviews in Fisheries Science, 4 (1996), 101-122. doi: doi:10.1080/10641269609388581.

[21]

P. L. Shafland and K. L. Foote, A reproducing population of Serrasalmus humeralis Valenciennes in southern Florida, Florida Scientist, 42 (1979), 206-214.

[22]

W. L. Shelton, Broodstock development for monosex production of grass carp, Aquaculture, 57 (1986), 311-319. doi: doi:10.1016/0044-8486(86)90209-7.

[23]

R. Temam, "Navier Stokes Equations and Nonlinear Functional Analysis," Society for Industrial and Applied Mathematics, Philadelphia, PA, 1985.

[24]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, NY, 1998.

[1]

Chuangxia Huang, Lihong Huang, Jianhong Wu. Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2427-2440. doi: 10.3934/dcdsb.2021138

[2]

Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099

[3]

Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101

[4]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure and Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288

[5]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451

[6]

Wei Feng, Xin Lu, Richard John Donovan Jr.. Population dynamics in a model for territory acquisition. Conference Publications, 2001, 2001 (Special) : 156-165. doi: 10.3934/proc.2001.2001.156

[7]

Sebastian Aniţa, Vincenzo Capasso, Ana-Maria Moşneagu. Global eradication for spatially structured populations by regional control. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2511-2533. doi: 10.3934/dcdsb.2018263

[8]

Surabhi Pandey, Ezio Venturino. A TB model: Is disease eradication possible in India?. Mathematical Biosciences & Engineering, 2018, 15 (1) : 233-254. doi: 10.3934/mbe.2018010

[9]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635

[10]

Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781

[11]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038

[12]

Chaohong Pan, Hongyong Wang, Chunhua Ou. Invasive speed for a competition-diffusion system with three species. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3515-3532. doi: 10.3934/dcdsb.2021194

[13]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks and Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[14]

Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643

[15]

Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395

[16]

Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

[17]

Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591

[18]

Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629

[19]

G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517

[20]

Ting-Hui Yang, Weinian Zhang, Kaijen Cheng. Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2867-2881. doi: 10.3934/dcdsb.2016077

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]