On the Well Posedness and Refined Estimates for the Global Attractor of the TYC Model
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Research Article On the Well Posedness and Refined Estimates for the Global Attractor of the TYC Model Rana D. Parshad1 and Juan B. Gutierrez2 1 2
Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13676, USA Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA
Correspondence should be addressed to Rana D. Parshad, [email protected] Received 14 July 2010; Accepted 2 November 2010 Academic Editor: Sandro Salsa Copyright q 2010 R. D. Parshad and J. B. Gutierrez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Trojan Y Chromosome strategy TYC is a theoretical method for eradication of invasive species. It requires constant introduction of artificial individuals into a target population, causing a shift in the sex ratio that ultimately leads to local extinction. In this work we demonstrate the existence of a unique weak solution to the infinite dimensional TYC system. Furthermore, we obtain improved estimates on the upper bounds for the Hausdorff and fractal dimensions of the global attractor of the TYC system, via the use of weighted Sobolev spaces. These results confirm that the TYC eradication strategy is a sound theoretical method of eradication of invasive species in a spatial setting. It also provides a solid ground for experiments in silico and validates the use of the TYC strategy in vivo.
1. Introduction An exotic species is a species that resides outside its native habitat. When it causes some sort of measurable damage, it is often referred to as invasive. The recent globalization process has expedited the pace at which exotic species are introduced into new environments. Once established, these species can be extremely difficult to manage and almost impossible to eradicate 1, 2. Studies have indicated that the losses caused by invasive species could be as much as $120 billion/year by 2004 3. The effect of these invaders is thus devastating 4. Current approaches for controlling exotic fish species are limited to general chemical control methods applied to small water bodies and/or small isolated populations that kill native fish in addition to the target fish 5. For example, the piscicide Rotenone has been used to eradicate exotic fish, but at the expense of killing all the endogenous fish, making it necessary to restock native fish from other sources 1, 2. A genetic strategy to cause extinction of invasive species was proposed by Gutierrez and Teem 6. This strategy is relevant to species amenable to sex reversal and with an XY sexdetermination system, in which males are the heterogametic sex carrying one X chromosome
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Boundary Value Problems
and one Y chromosome, XY and females are the homogametic sex carrying two X chromosomes, XX. The strategy relies on the fact that variations in the sex chromosome number can be produced through genetic manipulation, fo
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