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

On some spectral problems arising in dynamic populations

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.
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. doi: 10.3934/dcdsb.2009.11.805. Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle,", Masson, (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. 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. 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. 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, 1936 (2008), 115. doi: 10.1007/978-3-540-78273-5_3. Google Scholar

[7]

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

[8]

A. Henrot and M. Pierre, "Variation et optimisation de formes,", Math\'ematiques et Applications, 48 (2005). doi: 10.1007/3-540-37689-5. Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators,", \textbf{132}, 132 (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, (2009), 22. 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. doi: 10.3934/dcdsb.2009.11.805. Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle,", Masson, (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. 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. 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. 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, 1936 (2008), 115. doi: 10.1007/978-3-540-78273-5_3. Google Scholar

[7]

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

[8]

A. Henrot and M. Pierre, "Variation et optimisation de formes,", Math\'ematiques et Applications, 48 (2005). doi: 10.1007/3-540-37689-5. Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators,", \textbf{132}, 132 (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, (2009), 22. Google Scholar

[1]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

[2]

Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703

[3]

Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397

[4]

Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547

[5]

Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693

[6]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[7]

Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 417-433. doi: 10.3934/dcdsb.2007.8.417

[8]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[9]

Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265

[10]

Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005

[11]

Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289

[12]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[13]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

[14]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

[15]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[16]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[17]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

[18]

Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303

[19]

Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92

[20]

Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 909-918. doi: 10.3934/dcdsb.2016.21.909

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]