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June  2019, 12(3): 551-571. doi: 10.3934/krm.2019022

Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts

1. 

Laboratoire de Géodésie, IGN-LAREG, Bâtiment Lamarck A et B, 35 rue Hélène Brion, 75013 Paris, France

2. 

Sorbonne Universités, Inria, UPMC Univ Paris 06, Mamba project-team, Laboratoire Jacques-Louis Lions, Paris, France

3. 

Wolfgang Pauli Institute, c/o Faculty of Mathematics of the University of Vienna, Vienna, Austria

4. 

Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France

* Corresponding author: Marie Doumic

Received  January 2018 Revised  June 2018 Published  February 2019

Fund Project: M.D. is supported by ERC Starting Grant SKIPPERAD (number 306321).
P.G. is supported by ANR project KIBORD, ANR-13-BS01-0004

We study the asymptotic behaviour of the following linear growth-fragmentation equation
$ \frac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{ \partial x} \big(x u(t,x)\big) + B(x) u(t,x) = 4 B(2x)u(t,2x), $
and prove that under fairly general assumptions on the division rate
$ B(x), $
its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypocoercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted
$ L^2 $
space, where well-posedness is obtained via semigroup analysis. We also propose a non-diffusive numerical scheme, able to capture the oscillations.
Citation: Étienne Bernard, Marie Doumic, Pierre Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. Kinetic & Related Models, 2019, 12 (3) : 551-571. doi: 10.3934/krm.2019022
References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Berlin, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[2]

O. Arino, Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics, SIAM Rev., 34 (1992), 445-476.  doi: 10.1137/1034086.  Google Scholar

[3]

D. BalaguéJ. A. Cañizo and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, 6 (2013), 219-243.  doi: 10.3934/krm.2013.6.219.  Google Scholar

[4]

J. Banasiak, On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.  doi: 10.1002/mma.301.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006.  Google Scholar

[6]

J. Banasiak and W. Lamb, The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth, Kinet. Relat. Models, 5 (2012), 223-236.  doi: 10.3934/krm.2012.5.223.  Google Scholar

[7]

J. BanasiakK. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y.  Google Scholar

[8]

G. I. Bell, Cell growth and division: Ⅲ. conditions for balanced exponential growth in a mathematical model, Biophys. J., 8 (1968), 431-444.  doi: 10.1016/S0006-3495(68)86498-7.  Google Scholar

[9]

G. I. Bell and E. C. Anderson, Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.  doi: 10.1016/S0006-3495(67)86592-5.  Google Scholar

[10]

E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. Google Scholar

[11]

J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.  doi: 10.1007/s10097-003-0055-3.  Google Scholar

[12]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61.  doi: 10.1017/apr.2016.41.  Google Scholar

[13]

M. J. CáceresJ. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl., 96 (2011), 334-362.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar

[14]

B. Cloez, Limit theorems for some branching measure-valued processes, Adv. in Appl. Probab., 49 (2017), 549-580.  doi: 10.1017/apr.2017.12.  Google Scholar

[15]

O. DiekmannH. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.  doi: 10.1007/BF00277748.  Google Scholar

[16]

M. Doumic and M. Escobedo, Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297.  doi: 10.3934/krm.2016.9.251.  Google Scholar

[17]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.  Google Scholar

[18]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[20]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.  doi: 10.1016/j.anihpc.2004.06.001.  Google Scholar

[21]

P. Gabriel and F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., 27 (2014), 74-78.  doi: 10.1016/j.aml.2013.08.001.  Google Scholar

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79–105.  Google Scholar

[23]

P. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.  doi: 10.4310/CMS.2017.v15.n2.a13.  Google Scholar

[24]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429.  doi: 10.1214/09-AAP622.  Google Scholar

[25]

A. J. Hall and G. C. Wake, Functional-differential equations determining steady size distributions for populations of cells growing exponentially, J. Austral. Math. Soc. Ser. B, 31 (1990), 434-453.  doi: 10.1017/S0334270000006779.  Google Scholar

[26]

H. J. A. M. Heijmans, An eigenvalue problem related to cell growth, J. Math. Anal. Appl., 111 (1985), 253-280.  doi: 10.1016/0022-247X(85)90215-X.  Google Scholar

[27]

P. LaurençotB. Niethammer and J. J. L. Velázquez, Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel, Kinet. Relat. Models, 11 (2018), 933-952.  doi: 10.3934/krm.2018037.  Google Scholar

[28]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.  doi: 10.4310/CMS.2009.v7.n2.a12.  Google Scholar

[29]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.  doi: 10.1016/j.crma.2004.03.006.  Google Scholar

[30]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260. doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[31]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

[32]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.  Google Scholar

[33]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[34]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177.  doi: 10.1016/j.jde.2004.10.018.  Google Scholar

[35]

J. Sinko and W. Streifer, A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.  doi: 10.2307/1934592.  Google Scholar

[36]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[37]

A. A. Zaidi, B. Van Brunt and G. C. Wake, Solutions to an advanced functional partial differential equation of the pantograph type, Proc. A., 471 (2015), 20140947, 15pp. doi: 10.1098/rspa.2014.0947.  Google Scholar

[38]

A. A. ZaidiB. van Brunt and G. C. Wake, A model for asymmetrical cell division, Math. Biosc. Eng., 12 (2015), 491-501.  doi: 10.3934/mbe.2015.12.491.  Google Scholar

show all references

References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Berlin, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[2]

O. Arino, Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics, SIAM Rev., 34 (1992), 445-476.  doi: 10.1137/1034086.  Google Scholar

[3]

D. BalaguéJ. A. Cañizo and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, 6 (2013), 219-243.  doi: 10.3934/krm.2013.6.219.  Google Scholar

[4]

J. Banasiak, On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.  doi: 10.1002/mma.301.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006.  Google Scholar

[6]

J. Banasiak and W. Lamb, The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth, Kinet. Relat. Models, 5 (2012), 223-236.  doi: 10.3934/krm.2012.5.223.  Google Scholar

[7]

J. BanasiakK. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y.  Google Scholar

[8]

G. I. Bell, Cell growth and division: Ⅲ. conditions for balanced exponential growth in a mathematical model, Biophys. J., 8 (1968), 431-444.  doi: 10.1016/S0006-3495(68)86498-7.  Google Scholar

[9]

G. I. Bell and E. C. Anderson, Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.  doi: 10.1016/S0006-3495(67)86592-5.  Google Scholar

[10]

E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. Google Scholar

[11]

J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.  doi: 10.1007/s10097-003-0055-3.  Google Scholar

[12]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61.  doi: 10.1017/apr.2016.41.  Google Scholar

[13]

M. J. CáceresJ. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl., 96 (2011), 334-362.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar

[14]

B. Cloez, Limit theorems for some branching measure-valued processes, Adv. in Appl. Probab., 49 (2017), 549-580.  doi: 10.1017/apr.2017.12.  Google Scholar

[15]

O. DiekmannH. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.  doi: 10.1007/BF00277748.  Google Scholar

[16]

M. Doumic and M. Escobedo, Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297.  doi: 10.3934/krm.2016.9.251.  Google Scholar

[17]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.  Google Scholar

[18]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[20]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.  doi: 10.1016/j.anihpc.2004.06.001.  Google Scholar

[21]

P. Gabriel and F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., 27 (2014), 74-78.  doi: 10.1016/j.aml.2013.08.001.  Google Scholar

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79–105.  Google Scholar

[23]

P. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.  doi: 10.4310/CMS.2017.v15.n2.a13.  Google Scholar

[24]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429.  doi: 10.1214/09-AAP622.  Google Scholar

[25]

A. J. Hall and G. C. Wake, Functional-differential equations determining steady size distributions for populations of cells growing exponentially, J. Austral. Math. Soc. Ser. B, 31 (1990), 434-453.  doi: 10.1017/S0334270000006779.  Google Scholar

[26]

H. J. A. M. Heijmans, An eigenvalue problem related to cell growth, J. Math. Anal. Appl., 111 (1985), 253-280.  doi: 10.1016/0022-247X(85)90215-X.  Google Scholar

[27]

P. LaurençotB. Niethammer and J. J. L. Velázquez, Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel, Kinet. Relat. Models, 11 (2018), 933-952.  doi: 10.3934/krm.2018037.  Google Scholar

[28]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.  doi: 10.4310/CMS.2009.v7.n2.a12.  Google Scholar

[29]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.  doi: 10.1016/j.crma.2004.03.006.  Google Scholar

[30]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260. doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[31]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

[32]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.  Google Scholar

[33]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[34]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177.  doi: 10.1016/j.jde.2004.10.018.  Google Scholar

[35]

J. Sinko and W. Streifer, A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.  doi: 10.2307/1934592.  Google Scholar

[36]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[37]

A. A. Zaidi, B. Van Brunt and G. C. Wake, Solutions to an advanced functional partial differential equation of the pantograph type, Proc. A., 471 (2015), 20140947, 15pp. doi: 10.1098/rspa.2014.0947.  Google Scholar

[38]

A. A. ZaidiB. van Brunt and G. C. Wake, A model for asymmetrical cell division, Math. Biosc. Eng., 12 (2015), 491-501.  doi: 10.3934/mbe.2015.12.491.  Google Scholar

Figure 1.  The real part for the three first eigenvectors $ {\mathcal U} _0,\, {\mathcal U} _1,\, {\mathcal U} _2 $ for $ B(x) = x^2 $. We see the oscillatory behaviour for $ {\mathcal U} _1 $ and $ {\mathcal U} _2 $
Figure 2.  Two different initial conditions

Left: peak in $ x = 2. $ Right: $ u^{\rm{in}} (x) = x^2\exp(-x^2/2) $.

Figure 3.  Time evolution of $ \max\limits_{x>0} u(t,x)e^{-t} $

Left: for the peak as initial condition. Right: for the smooth initial condition.

Figure 4.  Size distribution $ u(t,x)e^{-t} $ at five different times (each time is in a different grey). Left: for the peak as initial condition. Right: for the smooth initial condition
Figure 5.  Left: initial distribution (full blue line) and dominant eigenvector (doted red line), for $ B(x) = x^3 $. We see that the constant such that $ u^{\rm{in}}\leq {\mathcal U}_0 $ is very large. Right: time evolution of Error$ _{E_2^n} $ (doted red line) and Error Mean$ _{E_2^n} $ (full blue line), in a log scale for the ordinates
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