December  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

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.
Citation: Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations and 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, Lecture Notes in Mathematics 446, Springer, New York, (1975), 5-49.

[2]

R. A. Fisher, The advance of advantageous genes, Ann. of Eugen, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[3]

V. A. Kostitzin, Mathematical biology, Lecture Notes in Biomathematics, 22 (1978), 413-423.

[4]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236. 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, 1987), Lecture Notes in Control and Inform. Sci., 114, Springer, Berlin, (1989), 202-207. 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, New York, 1984. 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), 391-421. 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), 191-206. 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., (in print). Erratum: J. Math. Biol., 67 (2013), p. 737 doi: 10.1007/s00285-013-0673-7.

show all references

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, Lecture Notes in Mathematics 446, Springer, New York, (1975), 5-49.

[2]

R. A. Fisher, The advance of advantageous genes, Ann. of Eugen, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[3]

V. A. Kostitzin, Mathematical biology, Lecture Notes in Biomathematics, 22 (1978), 413-423.

[4]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236. 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, 1987), Lecture Notes in Control and Inform. Sci., 114, Springer, Berlin, (1989), 202-207. 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, New York, 1984. 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), 391-421. 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), 191-206. 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., (in print). Erratum: J. Math. Biol., 67 (2013), p. 737 doi: 10.1007/s00285-013-0673-7.

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