Stabilization for a periodic predator-prey system. (English) Zbl 1163.35318

Summary: A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.


35B35 Stability in context of PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
Full Text: DOI EuDML


[1] W. Chen and M. Wang, “Qualitative analysis of predator-prey models with Beddington-De Angelis functional response and diffusion,” Mathematical and Computer Modelling, vol. 42, no. 1-2, pp. 31-44, 2005. · Zbl 1087.35053
[2] J.-D. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2002. · Zbl 1006.92001
[3] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, USA, 2003. · Zbl 1054.92042
[4] J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82-95, 1977. · Zbl 0348.34031
[5] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1967. · Zbl 0153.13602
[6] B. Ainseba, F. Heiser, and M. Langlais, “A mathematical analysis of a predator-prey system in a highly heterogeneous environment,” Differential and Integral Equations, vol. 15, no. 4, pp. 385-404, 2002. · Zbl 1011.35075
[7] B. Ainseba and S. Ani\cta, “Internal stabilizability for a reaction-diffusion problem modeling a predator-prey system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 4, pp. 491-501, 2005. · Zbl 1072.35090
[8] S. Ani\cta and M. Langlais, “Stabilization strategies for some reaction-diffusion systems,” submitted to, Nonlinear Analysis: Real World Applications, doi:10.1016/j.nonrwa.2007.09.003.
[9] F. Courchamp and G. Sugihara, “Modelling the biological control of an alien predator to protect island species from extinction,” Ecological Application, vol. 9, no. 1, pp. 112-123, 1999.
[10] V. Barbu, Partial Differential Equations and Boundary Value Problems, vol. 441 of Mathematics and Its Applications, Kluwer, Dordrecht, The Netherlands, 1998. · Zbl 0898.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.