Geodesic
Stabilization of a Predator-Prey System with Nonlocal Terms
S. Aniţa1
1 Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi 700506, Romania “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania
Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 6-16.

Voir la notice de l'article provenant de la source EDP Sciences

We investigate the zero-stabilizability for the prey population in a predator-prey system via a control which acts in a subregion ω of the habitat Ω, and on the predators only. The dynamics of both interacting populations is described by a reaction-diffusion system with nonlocal terms describing migrations. A necessary condition and a sufficient condition for the zero-stabilizability of the prey population are derived in terms of the sign of the principal eigenvalues to certain non-selfadjoint operators. In case of stabilizability, a constant stabilizing control is indicated. The rate of stabilization corresponding to such a stabilizing control is dictated by the principal eigenvalue of a certain operator. A large principal eigenvalue leads to a fast stabilization to zero of the prey population. A method to approximate all these principal eigenvalues is presented. Some final comments concerning the relationship between the stabilization rate and the properties of ω and Ω are given as well.
DOI : 10.1051/mmnp/201510602

S. Aniţa 1

1 Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi 700506, Romania “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania 
@article{MMNP_2015_10_6_a1,
     author = {S. Ani\c{t}a},
     title = {Stabilization of a {Predator-Prey} {System} with {Nonlocal} {Terms}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {6--16},
     publisher = {mathdoc},
     volume = {10},
     number = {6},
     year = {2015},
     doi = {10.1051/mmnp/201510602},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510602/}
}
TY  - JOUR
AU  - S. Aniţa
TI  - Stabilization of a Predator-Prey System with Nonlocal Terms
JO  - Mathematical modelling of natural phenomena
PY  - 2015
SP  - 6
EP  - 16
VL  - 10
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510602/
DO  - 10.1051/mmnp/201510602
LA  - en
ID  - MMNP_2015_10_6_a1
ER  - 
%0 Journal Article
%A S. Aniţa
%T Stabilization of a Predator-Prey System with Nonlocal Terms
%J Mathematical modelling of natural phenomena
%D 2015
%P 6-16
%V 10
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510602/
%R 10.1051/mmnp/201510602
%G en
%F MMNP_2015_10_6_a1
S. Aniţa. Stabilization of a Predator-Prey System with Nonlocal Terms. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 6-16. doi : 10.1051/mmnp/201510602. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510602/

[1] B. Ainseba, S. Aniţa Comm. Pure Appl. Anal. 2008 491 512

[2] L.-I. Aniţa, S. Aniţa, V. Arnăutu Appl. Math. Comput. 2013 10231 10244

[3] S. Aniţa Math. Model. Nat. Phenom. 2014 6 19

[4] S. Aniţa, V. Capasso Nonlin. Anal. Real World Appl. 2009 2026 2035

[5] S. Aniţa, W. Fitzgibbon, M. Langlais Discrete Cont. Dyn. Syst.- B 2009 805 822

[6] V. Barbu. Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston, 1993.

[7] V. Barbu. Partial Differential Equations and Boundary value problems. Kluwer Academic Press, Dordrecht, 1998.

[8] V. Capasso, R.E. Wilson SIAM J. Appl. Math. 1997 327 346

[9] K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.

[10] S. Genieys, V. Volpert, P. Auger Math. Modelling Nat. Phenom. 2006 65 82

[11] A. Henrot, El Haj Laamri, D. Schmitt Comm. Pure Appl. Anal. 2012 2429 2443

[12] B. Kawohl. Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150, 1985.

[13] J.-L. Lions. Controlabilité Exacte, Stabilisation et Perturbation de Systemes Distribués. RMA 8, Masson, Paris, 1988.

[14] A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980.

[15] J.D. Murray. Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edition. Springer-Verlag, New York, 2003.

[16] M.H. Protter, H.F. Weinberger. Maximum Principles in Differential Equations. Springer-Verlag, New York, 1984.

[17] J. Smoller. Shock Waves and Reaction Diffusion Equations. Springer Verlag, Berlin, 1983.

[18] V. Volpert, V. Vougalter. Stability and instability of solutions of a nonlocal reaction-diffusion equation when the essential spectrum crosses the imaginary axis. Preprint (2014), https://www.ma.utexas.edu/mparc/index − 14.html.

Cité par Sources :