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doi: 10.3934/dcdsb.2020169

Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures

Azmy S. Ackleh 1,, and  Nicolas Saintier 2,

1. 

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA
2. 

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Pabellón I - Ciudad Universitaria, Buenos Aires - Argentina

* Corresponding author: azmy.ackleh@louisiana.edu

Received  September 2019 Revised  February 2020 Published  May 2020

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on $ \mathbb{R}^d $. The selection-mutation kernel is described by a family of measures which allows the study of continuous and discrete kernels under the same setting. We rescale the selection-mutation kernel to obtain a diffusively rescaled selection-mutation model. We prove that if the rescaled selection-mutation kernel converges to a pure selection kernel then the solution of the diffusively rescaled model converges to a solution of an advection-diffusion equation.

Keywords: Selection-mutation equation, small mutation diffusive limit, nonlinear first-order hyperbolic equation on the space of measures.
Mathematics Subject Classification: Primary: 35L60, 35R06, 35Q92.
Citation: Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020169
References:
[1]

A. S. Ackleh, J. Cleveland and H. Thieme, Population dynamics under selection and mutation: Long-time behavior of differential equations in measure spaces, Journal of Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

[2]

A. S. Ackleh, B. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[3]

A. S. Ackleh and N. Saintier, Well-posedness for a system of transport and diffusion equations in measure spaces, Journal of Mathematical Analysis and Applications, in revision. Google Scholar

[4]

A. S. Ackleh, B. L. Ma and P. L. Salceanu, Persistence and global stability in a selection-mutation size-structured model, Journal of Biological Dynamics, 5 (2011), 436-453.  doi: 10.1080/17513758.2010.538729.  Google Scholar

[5]

L. Almeida, R. H. Chisholm, J. Clairambault, T. Lorenzi, A. Lorz and C Poucho, Why is evolution important in cancer and what mathematics should be used to treat cancer? Focus on drug resistance, Trends in Biomathematics: Modeling, Optimization and Computational Problems, Springer, Cham, (2018), 107–120. doi: 10.1007/978-3-319-91092-5_8.  Google Scholar

[6]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI, Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.  Google Scholar

[7]

P. Billingsley, Convergence of Probability Measures, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[8]

R. Burger and I. M. Bomze, Stationary distributions under mutation-selection balance: Structure and properties, Advances in Applied Probability, 28 (1996), 227-251.  doi: 10.2307/1427919.  Google Scholar

[9]

Á. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, Journal of Mathematical Biology, 48 (2004), 135-159.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[10]

Á. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotics of steady states of a selection mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 ((2013), 1123-1146.  doi: 10.1017/S0308210510001629.  Google Scholar

[11]

Á. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions, Journal of Mathematical Analysis and Applications, 444 (2016), 1515-1541.  doi: 10.1016/j.jmaa.2016.07.028.  Google Scholar

[12]

Á. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.  Google Scholar

[13]

J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[14]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[15]

N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69 (2006), 297-321.   Google Scholar

[16]

R. H. Chisholma, T. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical, treatment optimisation, Biochimica et Biophysica Acta, 1860 (2016). Google Scholar

[17]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-Posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002.  Google Scholar

[18]

R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics, Theoretical Population Biology, 67 (2005), 47-59.  doi: 10.1016/j.tpb.2004.08.001.  Google Scholar

[19]

R. M. Dudley, Convergence of Baire measures, Studia Mathematica, 27 (1966), 251-268.  doi: 10.4064/sm-27-3-251-268.  Google Scholar

[20]

J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[21]

G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, Journal of Statistical Physics, 81 (1995), 901-934.  doi: 10.1007/BF02179298.  Google Scholar

[22]

P. Gwiazda, A. Marciniak Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 105-138.  doi: 10.1007/s11117-017-0503-z.  Google Scholar

[23]

P. Gwiazda, T. Lorenz and A. Marciniak Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.  Google Scholar

[24]

S. C. Hille, T. Szarek, D. T. H. Worm and M. A. Ziemlanska, On a Schur-like property for spaces of measures, http://arXiv.org/pdf/1703.00677.pdf. Google Scholar

[25]

L. Hormander, The Analysis of Linear Partial Differential Operators. Ⅰ. Distribution Theory and Fourier Analysis, Second edition, Grundlehren der Mathematischen Wissenschaften, 256. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[26]

P. Magal and G. F. Webb, Mutation, selection, and recombination in a model of phenotype evolution, Discrete and Continuous Dynamical Systems, 6 (2000), 221-236.  doi: 10.3934/dcds.2000.6.221.  Google Scholar

[27]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations. Ⅱ: Chemotaxis equations, IAM Journal on Applied Mathematics, 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.  Google Scholar

[28]

B. Perthame, Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-19500-1.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

show all references

References:
[1]

A. S. Ackleh, J. Cleveland and H. Thieme, Population dynamics under selection and mutation: Long-time behavior of differential equations in measure spaces, Journal of Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

[2]

A. S. Ackleh, B. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[3]

A. S. Ackleh and N. Saintier, Well-posedness for a system of transport and diffusion equations in measure spaces, Journal of Mathematical Analysis and Applications, in revision. Google Scholar

[4]

A. S. Ackleh, B. L. Ma and P. L. Salceanu, Persistence and global stability in a selection-mutation size-structured model, Journal of Biological Dynamics, 5 (2011), 436-453.  doi: 10.1080/17513758.2010.538729.  Google Scholar

[5]

L. Almeida, R. H. Chisholm, J. Clairambault, T. Lorenzi, A. Lorz and C Poucho, Why is evolution important in cancer and what mathematics should be used to treat cancer? Focus on drug resistance, Trends in Biomathematics: Modeling, Optimization and Computational Problems, Springer, Cham, (2018), 107–120. doi: 10.1007/978-3-319-91092-5_8.  Google Scholar

[6]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI, Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.  Google Scholar

[7]

P. Billingsley, Convergence of Probability Measures, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[8]

R. Burger and I. M. Bomze, Stationary distributions under mutation-selection balance: Structure and properties, Advances in Applied Probability, 28 (1996), 227-251.  doi: 10.2307/1427919.  Google Scholar

[9]

Á. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, Journal of Mathematical Biology, 48 (2004), 135-159.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[10]

Á. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotics of steady states of a selection mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 ((2013), 1123-1146.  doi: 10.1017/S0308210510001629.  Google Scholar

[11]

Á. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions, Journal of Mathematical Analysis and Applications, 444 (2016), 1515-1541.  doi: 10.1016/j.jmaa.2016.07.028.  Google Scholar

[12]

Á. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.  Google Scholar

[13]

J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[14]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[15]

N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69 (2006), 297-321.   Google Scholar

[16]

R. H. Chisholma, T. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical, treatment optimisation, Biochimica et Biophysica Acta, 1860 (2016). Google Scholar

[17]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-Posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002.  Google Scholar

[18]

R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics, Theoretical Population Biology, 67 (2005), 47-59.  doi: 10.1016/j.tpb.2004.08.001.  Google Scholar

[19]

R. M. Dudley, Convergence of Baire measures, Studia Mathematica, 27 (1966), 251-268.  doi: 10.4064/sm-27-3-251-268.  Google Scholar

[20]

J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[21]

G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, Journal of Statistical Physics, 81 (1995), 901-934.  doi: 10.1007/BF02179298.  Google Scholar

[22]

P. Gwiazda, A. Marciniak Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 105-138.  doi: 10.1007/s11117-017-0503-z.  Google Scholar

[23]

P. Gwiazda, T. Lorenz and A. Marciniak Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.  Google Scholar

[24]

S. C. Hille, T. Szarek, D. T. H. Worm and M. A. Ziemlanska, On a Schur-like property for spaces of measures, http://arXiv.org/pdf/1703.00677.pdf. Google Scholar

[25]

L. Hormander, The Analysis of Linear Partial Differential Operators. Ⅰ. Distribution Theory and Fourier Analysis, Second edition, Grundlehren der Mathematischen Wissenschaften, 256. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[26]

P. Magal and G. F. Webb, Mutation, selection, and recombination in a model of phenotype evolution, Discrete and Continuous Dynamical Systems, 6 (2000), 221-236.  doi: 10.3934/dcds.2000.6.221.  Google Scholar

[27]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations. Ⅱ: Chemotaxis equations, IAM Journal on Applied Mathematics, 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.  Google Scholar

[28]

B. Perthame, Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-19500-1.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

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