On some spectral problems arising in dynamic populations

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November 2012, 11(6): 2429-2443. doi: 10.3934/cpaa.2012.11.2429

On some spectral problems arising in dynamic populations

Antoine Henrot ^1, , El-Haj Laamri ^2, and Didier Schmitt ^2,

1.	Institut Elie Cartan, Nancy Université - CNRS, B.P. 239, 54 506 Vandoeuvre-les-Nancy, France
2.	Institut Elie Cartan, Nancy Université - CNRS, B.P. 239, 54 506 Vandoeuvre-lès-Nancy, France, France

Received February 2011 Revised February 2011 Published April 2012

Abstract
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We study a spectral problem related to a reaction-diffusion model where preys and predators do not live on the same area. We are interested in the optimal zone where a control should take place. First, we prove existence of an optimal domain in a natural class. Then, it seems plausible that the optimal domain is localized in the intersection of the living areas of the two species. We prove this fact in one dimension for small sized domains.

Keywords: optimal location, predator-prey, Spectral problem.

Mathematics Subject Classification: Primary: 49J20; Secondary: 35P15, 92D25.

Citation: Antoine Henrot, El-Haj Laamri, Didier Schmitt. On some spectral problems arising in dynamic populations. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2429-2443. doi: 10.3934/cpaa.2012.11.2429

References:

[1]	S. Anita, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a reaction-diffusion systems posed on non coincident spatial domains, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 805-822. doi: 10.3934/dcdsb.2009.11.805.
[2]	H. Brézis, "Analyse Fonctionnelle," Masson, Paris, 1983.
[3]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity, J. Math. Anal. Appl., 338 (2008), 101-110. doi: 10.1016/j.jmaa.2007.05.011.
[4]	A. Ducrot, V. Guyonne and M. Langlais, Some Remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains, Discrete Contin. Dyn. Syst., 4 (2011), 67-82. doi: 10.3934/dcdss.2011.4.67.
[5]	H. Egnell, Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 1-48.
[6]	W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, In "Structured Population Models in Biology and Epidemiology'' (P. Magal and S. Ruan eds.), Lecture Notes in Mathematics (Mathematical Biosciences Sunseries) 1936, Springer, (2008), 115-164. doi: 10.1007/978-3-540-78273-5_3.
[7]	A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Frontiers in Mathematics, Birkhäuser, 2006. doi: 10.1007/3-7643-7706-2.
[8]	A. Henrot and M. Pierre, "Variation et optimisation de formes," Mathématiques et Applications, 48, Springer, 2005. doi: 10.1007/3-540-37689-5.
[9]	T. Kato, "Perturbation Theory for Linear Operators," 132, Springer-Verlag New-York Inc, 1966. doi: 10.1007/978-3-642-66282-9.
[10]	M. Langlais, Some mathematical reaction-diffusion problems arising in population dynamics and posed on non coincident spatial domains, Workshop "Partial Differential Equations and Applications" devoted to Michel Pierre's sixtieth birthday in Vittel, October 22-24, 2009.

show all references

References:

[1]	S. Anita, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a reaction-diffusion systems posed on non coincident spatial domains, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 805-822. doi: 10.3934/dcdsb.2009.11.805.
[2]	H. Brézis, "Analyse Fonctionnelle," Masson, Paris, 1983.
[3]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity, J. Math. Anal. Appl., 338 (2008), 101-110. doi: 10.1016/j.jmaa.2007.05.011.
[4]	A. Ducrot, V. Guyonne and M. Langlais, Some Remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains, Discrete Contin. Dyn. Syst., 4 (2011), 67-82. doi: 10.3934/dcdss.2011.4.67.
[5]	H. Egnell, Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 1-48.
[6]	W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, In "Structured Population Models in Biology and Epidemiology'' (P. Magal and S. Ruan eds.), Lecture Notes in Mathematics (Mathematical Biosciences Sunseries) 1936, Springer, (2008), 115-164. doi: 10.1007/978-3-540-78273-5_3.
[7]	A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Frontiers in Mathematics, Birkhäuser, 2006. doi: 10.1007/3-7643-7706-2.
[8]	A. Henrot and M. Pierre, "Variation et optimisation de formes," Mathématiques et Applications, 48, Springer, 2005. doi: 10.1007/3-540-37689-5.
[9]	T. Kato, "Perturbation Theory for Linear Operators," 132, Springer-Verlag New-York Inc, 1966. doi: 10.1007/978-3-642-66282-9.
[10]	M. Langlais, Some mathematical reaction-diffusion problems arising in population dynamics and posed on non coincident spatial domains, Workshop "Partial Differential Equations and Applications" devoted to Michel Pierre's sixtieth birthday in Vittel, October 22-24, 2009.

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