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Approximation methods for the distributed order calculus using the convolution quadrature
Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
R. M. Dudley,
Convergence of Baire measures, Studia Mathematica, 27 (1966), 251-268.
doi: 10.4064/sm-27-3-251-268. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
R. M. Dudley,
Convergence of Baire measures, Studia Mathematica, 27 (1966), 251-268.
doi: 10.4064/sm-27-3-251-268. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
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