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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

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 & 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.  Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle," Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Frontiers in Mathematics, Birkhäuser, 2006. doi: 10.1007/3-7643-7706-2.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et optimisation de formes," Mathématiques et Applications, 48, Springer, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," 132, Springer-Verlag New-York Inc, 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle," Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Frontiers in Mathematics, Birkhäuser, 2006. doi: 10.1007/3-7643-7706-2.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et optimisation de formes," Mathématiques et Applications, 48, Springer, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," 132, Springer-Verlag New-York Inc, 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[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. Google Scholar

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