The approximate controllability of a model for mutant selection

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2013, 2(4): 741-747. doi: 10.3934/eect.2013.2.741

The approximate controllability of a model for mutant selection

1.	School of Mathematics, University of Minnesota, 206 Church Street, Minneapolis, MN 55455

Received October 2012 Revised December 2012 Published November 2013

Abstract
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It is shown that the problem of eliminating a less-fit allele by allowing a mixture of genotypes whose densities satisfy a system of reaction-diffusion equations with population control to evolve in a reactor with impenetrable walls is approximately controllable.

Keywords: Parabolic system, approximate controllability., evolutionary selection, population genetics.

Mathematics Subject Classification: 35K45, 92D15, 93B0.

Citation: Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741

References:

[1]	D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation,, in Partial Differential Equations and Related Topics, 446* (1975), 5.*
[2]	R. A. Fisher, The advance of advantageous genes,, Ann. of Eugen, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[3]	V. A. Kostitzin, Mathematical biology,, Lecture Notes in Biomathematics, 22 (1978), 413.
[4]	W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rational Mech. Anal., 103* (1988), 193. doi: 10.1007/BF00251758.*
[5]	W. Littman and L. Markus, Remarks on exact controllability and stabilization of a hybrid system in elasticity through boundary damping,, Control of partial differential equations (Santiago de Compostela, (1989), 202. doi: 10.1007/BFb0002593.
[6]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Corrected reprint of the 1967 original. Springer-Verlag, (1967). doi: 10.1007/978-1-4612-5282-5.
[7]	P. Souplet and M. Winkler, The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection,, J. Math. Biol., 62 (2011), 391. doi: 10.1007/s00285-010-0339-7.
[8]	P. Souplet and M. Winkler, Classification of large-time behaviors in a rection-diffusion system modeling diallelic selection,, Math. Biosciences, 239 (2012), 191. doi: 10.1016/j.mbs.2012.05.005.
[9]	H. F. Weinberger, The retreat of the less fit allele in a population-controlled model for population genetics,, J. Math. Biol., 67 (2013). doi: 10.1007/s00285-013-0673-7.

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References:

[1]	D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation,, in Partial Differential Equations and Related Topics, 446* (1975), 5.*
[2]	R. A. Fisher, The advance of advantageous genes,, Ann. of Eugen, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[3]	V. A. Kostitzin, Mathematical biology,, Lecture Notes in Biomathematics, 22 (1978), 413.
[4]	W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rational Mech. Anal., 103* (1988), 193. doi: 10.1007/BF00251758.*
[5]	W. Littman and L. Markus, Remarks on exact controllability and stabilization of a hybrid system in elasticity through boundary damping,, Control of partial differential equations (Santiago de Compostela, (1989), 202. doi: 10.1007/BFb0002593.
[6]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Corrected reprint of the 1967 original. Springer-Verlag, (1967). doi: 10.1007/978-1-4612-5282-5.
[7]	P. Souplet and M. Winkler, The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection,, J. Math. Biol., 62 (2011), 391. doi: 10.1007/s00285-010-0339-7.
[8]	P. Souplet and M. Winkler, Classification of large-time behaviors in a rection-diffusion system modeling diallelic selection,, Math. Biosciences, 239 (2012), 191. doi: 10.1016/j.mbs.2012.05.005.
[9]	H. F. Weinberger, The retreat of the less fit allele in a population-controlled model for population genetics,, J. Math. Biol., 67 (2013). doi: 10.1007/s00285-013-0673-7.

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