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 & 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.

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, 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.

[1]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883

[2]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167

[3]

Zhilan Feng, Carlos Castillo-Chavez. The influence of infectious diseases on population genetics. Mathematical Biosciences & Engineering, 2006, 3 (3) : 467-483. doi: 10.3934/mbe.2006.3.467

[4]

Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821

[5]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[6]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263

[7]

Jinyuan Zhang, Aimin Zhou, Guixu Zhang, Hu Zhang. A clustering based mate selection for evolutionary optimization. Big Data & Information Analytics, 2017, 2 (1) : 77-85. doi: 10.3934/bdia.2017010

[8]

Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617

[9]

Yuan Lou, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 643-655. doi: 10.3934/dcds.2010.27.643

[10]

Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control & Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143

[11]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[12]

Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643

[13]

Yuanyuan Huang, Yiping Hao, Min Wang, Wen Zhou, Zhijun Wu. Optimality and stability of symmetric evolutionary games with applications in genetic selection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 503-523. doi: 10.3934/mbe.2015.12.503

[14]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[15]

Moncef Aouadi, Taoufik Moulahi. Approximate controllability of abstract nonsimple thermoelastic problem. Evolution Equations & Control Theory, 2015, 4 (4) : 373-389. doi: 10.3934/eect.2015.4.373

[16]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171

[17]

Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023

[18]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

[19]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

[20]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

2016 Impact Factor: 0.826

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]