# American Institute of Mathematical Sciences

November  2020, 25(11): 4127-4164. doi: 10.3934/dcdsb.2020091

## Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case

 1 Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany 2 Present address: Infection Analytics Program, Kirby Institute, UNSW Sydney, Wallace Wurth Building, High St, Kensington NSW 2052, Australia 2 Institute of Computational Biology, HelmholtzZentrum München - German Research Center for, Environmental Health, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany

Received  May 2019 Revised  November 2019 Published  April 2020

Fund Project: This work is part of the published dissertation thesis "Transport equations and plasmid-induced cellular heterogeneity" by ES (https://mediatum.ub.tum.de/1469742?id=1469742).
This work was funded by the German Research Foundation (DFG) priority program SPP1617 "Phenotypic heterogeneity and sociobiology of bacterial populations" (DFG MU 2339/2-2).

We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.

Citation: Eva Stadler, Johannes Müller. Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4127-4164. doi: 10.3934/dcdsb.2020091
##### References:
 [1] O. Arino, A survey of structured cell population dynamics, Acta Biotheor., 43 (1995), 3-25.  doi: 10.1007/BF00709430.  Google Scholar [2] K. B. Athreya and P. E. Ney, Branching Processes, Die Grundlehren der mathematischen Wissenschaften, vol. 196, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65371-1.  Google Scholar [3] T. Beebee and G. Rowe, An Introduction to Molecular Ecology, 2nd edition, Oxford University Press, Oxford University Press, 2008. Google Scholar [4] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rend. Circ. Mat. Palermo (2), 7 (1958), 261-269.  Google Scholar [5] W. E. Bentley, N. Mirjalili, D. C. Andersen, R. H. Davis and D. S. Kompala, Plasmid-encoded protein: The principal factor in the "metabolic burden" associated with recombinant bacteria, Biotechnol. Bioeng., 35 (1990), 668-681.  doi: 10.1002/bit.260350704.  Google Scholar [6] V. I. Bogachev, Measure Theory, vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [7] À. Calsina and J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.  Google Scholar [8] V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem: Application to fitness analysis, J. Math. Pures Appl., 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004.  Google Scholar [9] F. Campillo, N. Champagnat and C. Fritsch, Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models, J. Math. Biol., 73 (2016), 1781-1821.  doi: 10.1007/s00285-016-1012-6.  Google Scholar [10] N. Casali and A. Preston, E. coli Plasmid Vectors: Methods and Applications, Methods in Molecular Biology, vol. 235, Humana Press, Totowa, NJ, 2003. doi: 10.1385/1592594093.  Google Scholar [11] D. P. Clark and N. J. Pazdernik, Biotechnology, 2$^nd$ edition, Elsevier AP Cell Press, Amsterdam, 2015. Google Scholar [12] J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar [13] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-3-642-65690-3.  Google Scholar [14] M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.  Google Scholar [15] M. Doumic Jauffret 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 [16] S. M. Focardi and F. J. Fabozzi, The Mathematics of Financial Modeling and Investment Management, 1$^st$ edition, Frank J. Fabozzi Series, John Wiley & Sons, 2004. Google Scholar [17] V. V. Ganusov, A. V. Bril'kov and N. S. Pechurkin, Mathematical modeling of population dynamics of unstable plasmid-bearing bacteria strains during continuous cultivation in a chemostat, Biofizika, 45 (2000), 908-914.   Google Scholar [18] L. M. Graves, The Theory of Functions of Real Variables, 2$^nd$ edition, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956.  Google Scholar [19] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 3$^rd$ edition, Graduate studies in mathematics, vol. 40, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/040.  Google Scholar [20] H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a dell population, in The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., vol. 68, Springer, Berlin, 1986,185–202. doi: 10.1007/978-3-662-13159-6_5.  Google Scholar [21] H. Kuo and J. D. Keasling, A Monte Carlo simulation of plasmid replication during the bacterial division cycle, Biotechnol. Bioeng., 52 (1996), 633-647.  doi: 10.1002/(SICI)1097-0290(19961220)52:6<633::AID-BIT1>3.0.CO;2-P.  Google Scholar [22] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer, Berlin, Heidelberg, 1986. Google Scholar [24] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153.  doi: 10.1142/S0218202506001480.  Google Scholar [25] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar [26] S. Million-Weaver and M. Camps, Mechanisms of plasmid segregation: Have multicopy plasmids been overlooked?, Plasmid, 75 (2014), 27-36.  doi: 10.1016/j.plasmid.2014.07.002.  Google Scholar [27] 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 [28] J. Müller, K. Münch, B. Koopmann, E. Stadler, L. Roselius, D. Jahn and R. Münch, Plasmid segregation and accumulation, preprint, arXiv: 1701.03448. Google Scholar [29] K. M. Münch, J. Müller, S. Wienecke, S. Bergmann, S. Heyber, R. Biedendieck and et al., Polar fixation of plasmids during recombinant protein production in bacillus megaterium results in population heterogeneity, Applied Environmental Microbiology, 81 (2015), 5976-5986.  doi: 10.1128/AEM.00807-15.  Google Scholar [30] R. P. Novick, R. C. Clowes, S. N. Cohen, R. Curtiss, N. Datta and S. Falkow, Uniform nomenclature for bacterial plasmids: A proposal, Microbiology and Molecular Biology Reviews, 40 (1976), 168-189.  doi: 10.1128/MMBR.40.1.168-189.1976.  Google Scholar [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [32] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4.  Google Scholar [33] J. Pogliano, T. Q. Ho, Z. Zhong and D. R. Helinski, Multicopy plasmids are clustered and localized in Escherichia coli, PNAS, 98 (2001), 4486-4491.  doi: 10.1073/pnas.081075798.  Google Scholar [34] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2018. Google Scholar [35] S. Srivastava, Genetics of Bacteria, Springer, India, 2013. doi: 10.1007/978-81-322-1090-0.  Google Scholar [36] E. Stadler, Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids, J. Math. Biol., 78 (2019), 1299-1330.  doi: 10.1007/s00285-018-1310-2.  Google Scholar [37] F. M. Stewart and B. R. Levin, The population biology of bacterial plasmids: A priori conditions for the existence of conjugationally transmitted factors, Genetics, 87 (1977), 209-228.   Google Scholar [38] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.  Google Scholar [39] D. G. Zill and P. Shanahan, A First Course in Complex Analysis with Applications, Jones and Bartlett, Boston, 2003. Google Scholar

show all references

##### References:
 [1] O. Arino, A survey of structured cell population dynamics, Acta Biotheor., 43 (1995), 3-25.  doi: 10.1007/BF00709430.  Google Scholar [2] K. B. Athreya and P. E. Ney, Branching Processes, Die Grundlehren der mathematischen Wissenschaften, vol. 196, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65371-1.  Google Scholar [3] T. Beebee and G. Rowe, An Introduction to Molecular Ecology, 2nd edition, Oxford University Press, Oxford University Press, 2008. Google Scholar [4] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rend. Circ. Mat. Palermo (2), 7 (1958), 261-269.  Google Scholar [5] W. E. Bentley, N. Mirjalili, D. C. Andersen, R. H. Davis and D. S. Kompala, Plasmid-encoded protein: The principal factor in the "metabolic burden" associated with recombinant bacteria, Biotechnol. Bioeng., 35 (1990), 668-681.  doi: 10.1002/bit.260350704.  Google Scholar [6] V. I. Bogachev, Measure Theory, vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [7] À. Calsina and J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.  Google Scholar [8] V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem: Application to fitness analysis, J. Math. Pures Appl., 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004.  Google Scholar [9] F. Campillo, N. Champagnat and C. Fritsch, Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models, J. Math. Biol., 73 (2016), 1781-1821.  doi: 10.1007/s00285-016-1012-6.  Google Scholar [10] N. Casali and A. Preston, E. coli Plasmid Vectors: Methods and Applications, Methods in Molecular Biology, vol. 235, Humana Press, Totowa, NJ, 2003. doi: 10.1385/1592594093.  Google Scholar [11] D. P. Clark and N. J. Pazdernik, Biotechnology, 2$^nd$ edition, Elsevier AP Cell Press, Amsterdam, 2015. Google Scholar [12] J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar [13] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-3-642-65690-3.  Google Scholar [14] M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.  Google Scholar [15] M. Doumic Jauffret 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 [16] S. M. Focardi and F. J. Fabozzi, The Mathematics of Financial Modeling and Investment Management, 1$^st$ edition, Frank J. Fabozzi Series, John Wiley & Sons, 2004. Google Scholar [17] V. V. Ganusov, A. V. Bril'kov and N. S. Pechurkin, Mathematical modeling of population dynamics of unstable plasmid-bearing bacteria strains during continuous cultivation in a chemostat, Biofizika, 45 (2000), 908-914.   Google Scholar [18] L. M. Graves, The Theory of Functions of Real Variables, 2$^nd$ edition, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956.  Google Scholar [19] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 3$^rd$ edition, Graduate studies in mathematics, vol. 40, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/040.  Google Scholar [20] H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a dell population, in The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., vol. 68, Springer, Berlin, 1986,185–202. doi: 10.1007/978-3-662-13159-6_5.  Google Scholar [21] H. Kuo and J. D. Keasling, A Monte Carlo simulation of plasmid replication during the bacterial division cycle, Biotechnol. Bioeng., 52 (1996), 633-647.  doi: 10.1002/(SICI)1097-0290(19961220)52:6<633::AID-BIT1>3.0.CO;2-P.  Google Scholar [22] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer, Berlin, Heidelberg, 1986. Google Scholar [24] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153.  doi: 10.1142/S0218202506001480.  Google Scholar [25] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar [26] S. Million-Weaver and M. Camps, Mechanisms of plasmid segregation: Have multicopy plasmids been overlooked?, Plasmid, 75 (2014), 27-36.  doi: 10.1016/j.plasmid.2014.07.002.  Google Scholar [27] 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 [28] J. Müller, K. Münch, B. Koopmann, E. Stadler, L. Roselius, D. Jahn and R. Münch, Plasmid segregation and accumulation, preprint, arXiv: 1701.03448. Google Scholar [29] K. M. Münch, J. Müller, S. Wienecke, S. Bergmann, S. Heyber, R. Biedendieck and et al., Polar fixation of plasmids during recombinant protein production in bacillus megaterium results in population heterogeneity, Applied Environmental Microbiology, 81 (2015), 5976-5986.  doi: 10.1128/AEM.00807-15.  Google Scholar [30] R. P. Novick, R. C. Clowes, S. N. Cohen, R. Curtiss, N. Datta and S. Falkow, Uniform nomenclature for bacterial plasmids: A proposal, Microbiology and Molecular Biology Reviews, 40 (1976), 168-189.  doi: 10.1128/MMBR.40.1.168-189.1976.  Google Scholar [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [32] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4.  Google Scholar [33] J. Pogliano, T. Q. Ho, Z. Zhong and D. R. Helinski, Multicopy plasmids are clustered and localized in Escherichia coli, PNAS, 98 (2001), 4486-4491.  doi: 10.1073/pnas.081075798.  Google Scholar [34] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2018. Google Scholar [35] S. Srivastava, Genetics of Bacteria, Springer, India, 2013. doi: 10.1007/978-81-322-1090-0.  Google Scholar [36] E. Stadler, Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids, J. Math. Biol., 78 (2019), 1299-1330.  doi: 10.1007/s00285-018-1310-2.  Google Scholar [37] F. M. Stewart and B. R. Levin, The population biology of bacterial plasmids: A priori conditions for the existence of conjugationally transmitted factors, Genetics, 87 (1977), 209-228.   Google Scholar [38] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1.  Google Scholar [39] D. G. Zill and P. Shanahan, A First Course in Complex Analysis with Applications, Jones and Bartlett, Boston, 2003. Google Scholar
Numerically constructed eigenfunctions for $\Phi(\xi) = 6\,\xi\,(1-\xi)$, $\mu = 0.1/h$, $b(z) = z(1-z)/h$, and different $\beta$, viz. $\beta = 0.45/h$ (black), $0.5/h$ (dark gray), and $0.55/h$ (light gray). The different cell division rates lead to different behavior of the eigenfunction $\mathcal{U}(z)$ at the maximal plasmid number $z_0 = 1$. The eigenfunction was numerically constructed using the software R [34] as described in [36,Section 5]
 [1] Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223 [2] Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063 [3] Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020254 [4] Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040 [5] Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251 [6] Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043 [7] Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529 [8] Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563 [9] Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589 [10] Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 [11] José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models. Networks & Heterogeneous Media, 2020, 15 (3) : 369-387. doi: 10.3934/nhm.2020023 [12] Jaroslaw Smieja, Marzena Dolbniak. Sensitivity of signaling pathway dynamics to plasmid transfection and its consequences. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1207-1222. doi: 10.3934/mbe.2016039 [13] Mengli Hao, Ting Gao, Jinqiao Duan, Wei Xu. Non-Gaussian dynamics of a tumor growth system with immunization. Inverse Problems & Imaging, 2013, 7 (3) : 697-716. doi: 10.3934/ipi.2013.7.697 [14] Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67 [15] Danielle Hilhorst, Masato Iida, Masayasu Mimura, Hirokazu Ninomiya. Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 233-244. doi: 10.3934/dcds.2008.21.233 [16] Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729 [17] Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126 [18] Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701 [19] Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569 [20] Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. Journal of Geometric Mechanics, 2020, 12 (3) : 395-420. doi: 10.3934/jgm.2020022

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables