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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 and 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. [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. [3] D. Balagué, J. 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. [4] J. Banasiak, On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.  doi: 10.1002/mma.301. [5] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006. [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. [7] J. Banasiak, K. 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. [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. [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. [10] E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. [11] J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.  doi: 10.1007/s10097-003-0055-3. [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. [13] M. J. Cáceres, J. 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. [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. [15] O. Diekmann, H. 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. [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. [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. [18] M. Doumic, M. Hoffmann, N. 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. [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. [20] M. Escobedo, S. 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. [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. [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. [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. [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. [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. [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. [27] P. Laurençot, B. 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. [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. [29] P. Michel, S. 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. [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. [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. [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. [33] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [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. [35] J. Sinko and W. Streifer, A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.  doi: 10.2307/1934592. [36] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5. [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. [38] A. A. Zaidi, B. 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.

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. [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. [3] D. Balagué, J. 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. [4] J. Banasiak, On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.  doi: 10.1002/mma.301. [5] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006. [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. [7] J. Banasiak, K. 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. [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. [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. [10] E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. [11] J. Bertoin, The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.  doi: 10.1007/s10097-003-0055-3. [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. [13] M. J. Cáceres, J. 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. [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. [15] O. Diekmann, H. 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. [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. [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. [18] M. Doumic, M. Hoffmann, N. 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. [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. [20] M. Escobedo, S. 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. [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. [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. [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. [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. [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. [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. [27] P. Laurençot, B. 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. [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. [29] P. Michel, S. 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. [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. [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. [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. [33] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [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. [35] J. Sinko and W. Streifer, A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.  doi: 10.2307/1934592. [36] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5. [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. [38] A. A. Zaidi, B. 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.
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 ${\mathcal U} _2$
Two different initial conditions

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

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.

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
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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