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 & 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. doi: doi:10.1016/j.ecolmodel.2003.11.009.

[2]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley, (2003).

[3]

"Economic Research Service, United States Department of Agriculture,", Agricultural Outlook: Statistical Indicators, May 2009., Retrieved on 06/01/2009 from \url{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, (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. 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.

[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]

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. 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. 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. 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. 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. doi: doi:10.2307/3434923.

[14]

O. T. A., Harmful non-indigenous species in the United States,, OTA-F-565 U.S. Congress, (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.

[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.

[18]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge University Press, (2001).

[19]

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

[20]

P. L. Shafland, Exotic fishes of Florida 1994,, Reviews in Fisheries Science, 4 (1996), 101. 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.

[22]

W. L. Shelton, Broodstock development for monosex production of grass carp,, Aquaculture, 57 (1986), 311. 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, (1985).

[24]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Springer, (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. doi: doi:10.1016/j.ecolmodel.2003.11.009.

[2]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley, (2003).

[3]

"Economic Research Service, United States Department of Agriculture,", Agricultural Outlook: Statistical Indicators, May 2009., Retrieved on 06/01/2009 from \url{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, (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. 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.

[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]

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. 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. 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. 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. 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. doi: doi:10.2307/3434923.

[14]

O. T. A., Harmful non-indigenous species in the United States,, OTA-F-565 U.S. Congress, (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.

[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.

[18]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge University Press, (2001).

[19]

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

[20]

P. L. Shafland, Exotic fishes of Florida 1994,, Reviews in Fisheries Science, 4 (1996), 101. 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.

[22]

W. L. Shelton, Broodstock development for monosex production of grass carp,, Aquaculture, 57 (1986), 311. 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, (1985).

[24]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1998).

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