
-
Previous Article
BGK model of the multi-species Uehling-Uhlenbeck equation
- KRM Home
- This Issue
- Next Article
Kinetic modelling of colonies of myxobacteria
1. | University of Vienna, Faculty for Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria |
2. | University College London, Dept. of Mathematics, 25 Gordon Street, WC1H 0AY London, UK |
A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions.
For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented.
References:
[1] |
R. J. Alonso,
Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.
doi: 10.1512/iumj.2009.58.3506. |
[2] |
R. Alonso, V. Bagland, Y. Cheng and B. Lods,
One-dimensional dissipative Boltzmann equation: Measure solutions, cooling rate, and self-similar profile, SIAM J. Math. Anal., 50 (2018), 1278-1321.
doi: 10.1137/17M1136791. |
[3] |
R. J. Alonso and B. Lods,
Two proofs of Haff's law for dissipative gases: The use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.
doi: 10.1016/j.jmaa.2012.07.045. |
[4] |
I. S. Aranson and L. S. Tsimring, Pattern formation of microtubules and motors: Inelastic interaction of polar rods,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 71 (2005), 050901. Google Scholar |
[5] |
A. Baskaran and M. C. Marchetti, Enhanced diffusion and ordering of self-propelled rods,, Phys. Rev. Lett., 101 (2008), 268101.
doi: 10.1103/PhysRevLett.101.268101. |
[6] |
A. Baskaran and M. C. Marchetti, Nonequilibrium statistical mechanics of self propelled hard rods,, J. Stat. Mech., 2010 (2010), P04019.
doi: 10.1103/PhysRevE.77.011920. |
[7] |
D. Benedetto and M. Pulvirenti,
On the one-dimensional Boltzmann equation for granular flows, M2AN, 35 (2001), 899-905.
doi: 10.1051/m2an:2001141. |
[8] |
E. Ben-Naim and P. L. Krapivsky, Alignment of rods and partition of integers,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 73 (2006), 031109.
doi: 10.1103/PhysRevE.73.031109. |
[9] |
E. Bertin, M. Droz and G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2006), 445001.
doi: 10.1088/1751-8113/42/44/445001. |
[10] |
A. V. Bobylev, J. A.Carrillo and I. M. Gamba,
On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Stat. Phys., 98 (2000), 743-773.
doi: 10.1023/A:1018627625800. |
[11] |
A. V. Bobylev and C. Cercignani, Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions,, J. Stat. Phys., 110 (2003), 333–375.
doi: 10.1023/A:1021031031038. |
[12] |
L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, WTB Wissenschaftliche Taschenbücher book series, 68, In Kinetische Theorie II, pp 115-225
doi: 10.1007/978-3-322-84986-1_3. |
[13] |
E. Carlen, M. C. Carvalho, P. Degond and B. Wennberg,
A Boltzmann model for rod alignment and schooling fish, Nonlinearity, 28 (2015), 1783-1804.
doi: 10.1088/0951-7715/28/6/1783. |
[14] |
J. A. Carrillo and G. Toscani,
Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198.
|
[15] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[16] |
P. Degond, A. Frouvelle and G. Raoul,
Local stability of perfect alignment for a spatially homogeneous kinetic model, J. Stat. Phys., 157 (2014), 84-112.
doi: 10.1007/s10955-014-1062-3. |
[17] |
P. Degond, A. Manhart and H. Yu,
A continuum model of nematic alignment of self-propelled particles, DCDS-B, 22 (2017), 1295-1327.
doi: 10.3934/dcdsb.2017063. |
[18] |
P. Degond, A. Manhart and H. Yu,
An age-structured continuum model for myxobacteria, M3AS, 28 (2018), 1737-1770.
doi: 10.1142/S0218202518400043. |
[19] |
P. K. Haff,
Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech., 134 (1983), 401-30.
doi: 10.1017/S0022112083003419. |
[20] |
J. Hodgkin and D. Kaiser,
Genetics of gliding motility in Myxococcus xanthus (Myxobacterales): two gene systems control movement, Mol. Gen. Genet., 171 (1979), 177-191.
doi: 10.1007/BF00270004. |
[21] |
O. A. Igoshin, A. Mogilner, R. D. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling,, \emphPNAS, 98 (2001), 14913-14918.
doi: 10.1073/pnas.221579598. |
[22] |
O. A. Igoshin and G. Oster,
Rippling of myxobacteria, Math. Biosci., 188 (2004), 221-233.
doi: 10.1016/j.mbs.2003.04.001. |
[23] |
O. A. Igoshin, R. Welch, D. Kaiser and G. Oster,
Waves and aggregation patterns in myxobacteria, PNAS, 101 (2004), 4256-4261.
doi: 10.1073/pnas.0400704101. |
[24] |
P.-E. Jabin and T. Rey,
Hydrodynamic limit of granular gases to pressureless Euler in dimension 1, Quart. Appl. Math., 75 (2017), 155-179.
doi: 10.1090/qam/1442. |
[25] |
Y. Jiang, O. Sozinova and M. Alber,
On modeling complex collective behavior in myxobacteria, Adv. in Complex Syst., 9 (2006), 353-367.
doi: 10.1142/S0219525906000860. |
[26] |
L. Jelsbak and L. Sogaard-Andersen,
The cell surface-associated intercellular C-signal induces behavioral changes in individual Myxococcus xanthus cells during fruiting body morphogenesis, PNAS, 96 (1999), 5031-5036.
doi: 10.1073/pnas.96.9.5031. |
[27] |
S. K. Kim and D. Kaiser,
C-factor: A cell-cell signaling protein required for fruiting body morphogenesis of M. xanthus, Cell, 61 (1990), 19-26.
doi: 10.1016/0092-8674(90)90211-V. |
[28] |
O. E. Lanford,
Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111.
|
[29] |
E. M. F. Mauriello, T. Mignot, Z. Yang and D. R. Zusman,
Gliding motility revisited: How do the myxobacteria move without flagella?, Microbiol. Mol. Biol. Rev., 74 (2010), 229-249.
doi: 10.1128/MMBR.00043-09. |
[30] |
S. Mischler, C. Mouhot and M. Rodriguez Ricard,
Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.
doi: 10.1007/s10955-006-9096-9. |
[31] |
S. Mischler and C. Mouhot,
Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.
doi: 10.1007/s10955-006-9097-8. |
[32] |
B. Nan and D. R. Zusman,
Uncovering the mystery of gliding motility in the myxobacteria, Annu. Rev. Genet., 45 (2011), 21-39.
doi: 10.1146/annurev-genet-110410-132547. |
[33] |
B. Sager and D. Kaiser,
Intercellular C-signaling and the traveling waves of Myxococcus, Genes Dev., 8 (1994), 2793-2804.
doi: 10.1101/gad.8.23.2793. |
[34] |
G. Toscani, Hydrodynamics from the Dissipative Boltzmann Equation, , in: G. Capriz, P.M. Mariano, P. Giovine (eds), Mathematical Models of Granular Matter, Lect. Notes in Math. 1937, Springer, Berlin–Heidelberg, 2008.
doi: 10.1007/978-3-540-78277-3_3. |
[35] |
I. Tristani,
Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Functional Anal., 270 (2016), 1922-1970.
doi: 10.1016/j.jfa.2015.09.025. |
[36] |
C. Villani, Topics in Optimal Transportation, , Graduate Studies in Math. 58, AMS, 2003.
doi: 10.1090/gsm/058. |
[37] |
D. Wall and D. Kaiser,
Type Ⅳ pili and cell motility, Mol. Microbiol., 32 (1999), 1-10.
doi: 10.1046/j.1365-2958.1999.01339.x. |
[38] |
R. Welch and D. Kaiser,
Cell behavior in traveling wave patterns of myxobacteria, PNAS, 98 (2001), 14907-14912.
doi: 10.1073/pnas.261574598. |
[39] |
C. Wolgemuth, E. Hoiczyk, D. Kaiser and G. Oster,
How myxobacteria glide, Curr. Biol., 12 (2002), 369-377.
doi: 10.1016/S0960-9822(02)00716-9. |
show all references
References:
[1] |
R. J. Alonso,
Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.
doi: 10.1512/iumj.2009.58.3506. |
[2] |
R. Alonso, V. Bagland, Y. Cheng and B. Lods,
One-dimensional dissipative Boltzmann equation: Measure solutions, cooling rate, and self-similar profile, SIAM J. Math. Anal., 50 (2018), 1278-1321.
doi: 10.1137/17M1136791. |
[3] |
R. J. Alonso and B. Lods,
Two proofs of Haff's law for dissipative gases: The use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.
doi: 10.1016/j.jmaa.2012.07.045. |
[4] |
I. S. Aranson and L. S. Tsimring, Pattern formation of microtubules and motors: Inelastic interaction of polar rods,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 71 (2005), 050901. Google Scholar |
[5] |
A. Baskaran and M. C. Marchetti, Enhanced diffusion and ordering of self-propelled rods,, Phys. Rev. Lett., 101 (2008), 268101.
doi: 10.1103/PhysRevLett.101.268101. |
[6] |
A. Baskaran and M. C. Marchetti, Nonequilibrium statistical mechanics of self propelled hard rods,, J. Stat. Mech., 2010 (2010), P04019.
doi: 10.1103/PhysRevE.77.011920. |
[7] |
D. Benedetto and M. Pulvirenti,
On the one-dimensional Boltzmann equation for granular flows, M2AN, 35 (2001), 899-905.
doi: 10.1051/m2an:2001141. |
[8] |
E. Ben-Naim and P. L. Krapivsky, Alignment of rods and partition of integers,, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 73 (2006), 031109.
doi: 10.1103/PhysRevE.73.031109. |
[9] |
E. Bertin, M. Droz and G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2006), 445001.
doi: 10.1088/1751-8113/42/44/445001. |
[10] |
A. V. Bobylev, J. A.Carrillo and I. M. Gamba,
On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions, J. Stat. Phys., 98 (2000), 743-773.
doi: 10.1023/A:1018627625800. |
[11] |
A. V. Bobylev and C. Cercignani, Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions,, J. Stat. Phys., 110 (2003), 333–375.
doi: 10.1023/A:1021031031038. |
[12] |
L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, WTB Wissenschaftliche Taschenbücher book series, 68, In Kinetische Theorie II, pp 115-225
doi: 10.1007/978-3-322-84986-1_3. |
[13] |
E. Carlen, M. C. Carvalho, P. Degond and B. Wennberg,
A Boltzmann model for rod alignment and schooling fish, Nonlinearity, 28 (2015), 1783-1804.
doi: 10.1088/0951-7715/28/6/1783. |
[14] |
J. A. Carrillo and G. Toscani,
Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198.
|
[15] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[16] |
P. Degond, A. Frouvelle and G. Raoul,
Local stability of perfect alignment for a spatially homogeneous kinetic model, J. Stat. Phys., 157 (2014), 84-112.
doi: 10.1007/s10955-014-1062-3. |
[17] |
P. Degond, A. Manhart and H. Yu,
A continuum model of nematic alignment of self-propelled particles, DCDS-B, 22 (2017), 1295-1327.
doi: 10.3934/dcdsb.2017063. |
[18] |
P. Degond, A. Manhart and H. Yu,
An age-structured continuum model for myxobacteria, M3AS, 28 (2018), 1737-1770.
doi: 10.1142/S0218202518400043. |
[19] |
P. K. Haff,
Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech., 134 (1983), 401-30.
doi: 10.1017/S0022112083003419. |
[20] |
J. Hodgkin and D. Kaiser,
Genetics of gliding motility in Myxococcus xanthus (Myxobacterales): two gene systems control movement, Mol. Gen. Genet., 171 (1979), 177-191.
doi: 10.1007/BF00270004. |
[21] |
O. A. Igoshin, A. Mogilner, R. D. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling,, \emphPNAS, 98 (2001), 14913-14918.
doi: 10.1073/pnas.221579598. |
[22] |
O. A. Igoshin and G. Oster,
Rippling of myxobacteria, Math. Biosci., 188 (2004), 221-233.
doi: 10.1016/j.mbs.2003.04.001. |
[23] |
O. A. Igoshin, R. Welch, D. Kaiser and G. Oster,
Waves and aggregation patterns in myxobacteria, PNAS, 101 (2004), 4256-4261.
doi: 10.1073/pnas.0400704101. |
[24] |
P.-E. Jabin and T. Rey,
Hydrodynamic limit of granular gases to pressureless Euler in dimension 1, Quart. Appl. Math., 75 (2017), 155-179.
doi: 10.1090/qam/1442. |
[25] |
Y. Jiang, O. Sozinova and M. Alber,
On modeling complex collective behavior in myxobacteria, Adv. in Complex Syst., 9 (2006), 353-367.
doi: 10.1142/S0219525906000860. |
[26] |
L. Jelsbak and L. Sogaard-Andersen,
The cell surface-associated intercellular C-signal induces behavioral changes in individual Myxococcus xanthus cells during fruiting body morphogenesis, PNAS, 96 (1999), 5031-5036.
doi: 10.1073/pnas.96.9.5031. |
[27] |
S. K. Kim and D. Kaiser,
C-factor: A cell-cell signaling protein required for fruiting body morphogenesis of M. xanthus, Cell, 61 (1990), 19-26.
doi: 10.1016/0092-8674(90)90211-V. |
[28] |
O. E. Lanford,
Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111.
|
[29] |
E. M. F. Mauriello, T. Mignot, Z. Yang and D. R. Zusman,
Gliding motility revisited: How do the myxobacteria move without flagella?, Microbiol. Mol. Biol. Rev., 74 (2010), 229-249.
doi: 10.1128/MMBR.00043-09. |
[30] |
S. Mischler, C. Mouhot and M. Rodriguez Ricard,
Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅰ: The Cauchy problem, J. Stat. Phys., 124 (2006), 655-702.
doi: 10.1007/s10955-006-9096-9. |
[31] |
S. Mischler and C. Mouhot,
Cooling process for inelastic Boltzmann equations for hard spheres, Part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.
doi: 10.1007/s10955-006-9097-8. |
[32] |
B. Nan and D. R. Zusman,
Uncovering the mystery of gliding motility in the myxobacteria, Annu. Rev. Genet., 45 (2011), 21-39.
doi: 10.1146/annurev-genet-110410-132547. |
[33] |
B. Sager and D. Kaiser,
Intercellular C-signaling and the traveling waves of Myxococcus, Genes Dev., 8 (1994), 2793-2804.
doi: 10.1101/gad.8.23.2793. |
[34] |
G. Toscani, Hydrodynamics from the Dissipative Boltzmann Equation, , in: G. Capriz, P.M. Mariano, P. Giovine (eds), Mathematical Models of Granular Matter, Lect. Notes in Math. 1937, Springer, Berlin–Heidelberg, 2008.
doi: 10.1007/978-3-540-78277-3_3. |
[35] |
I. Tristani,
Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Functional Anal., 270 (2016), 1922-1970.
doi: 10.1016/j.jfa.2015.09.025. |
[36] |
C. Villani, Topics in Optimal Transportation, , Graduate Studies in Math. 58, AMS, 2003.
doi: 10.1090/gsm/058. |
[37] |
D. Wall and D. Kaiser,
Type Ⅳ pili and cell motility, Mol. Microbiol., 32 (1999), 1-10.
doi: 10.1046/j.1365-2958.1999.01339.x. |
[38] |
R. Welch and D. Kaiser,
Cell behavior in traveling wave patterns of myxobacteria, PNAS, 98 (2001), 14907-14912.
doi: 10.1073/pnas.261574598. |
[39] |
C. Wolgemuth, E. Hoiczyk, D. Kaiser and G. Oster,
How myxobacteria glide, Curr. Biol., 12 (2002), 369-377.
doi: 10.1016/S0960-9822(02)00716-9. |






[1] |
Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 |
[2] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[3] |
Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365 |
[4] |
Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 |
[5] |
Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 |
[6] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[7] |
Raz Kupferman, Cy Maor. The emergence of torsion in the continuum limit of distributed edge-dislocations. Journal of Geometric Mechanics, 2015, 7 (3) : 361-387. doi: 10.3934/jgm.2015.7.361 |
[8] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[9] |
Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 |
[10] |
Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161 |
[11] |
Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 |
[12] |
Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427 |
[13] |
Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 |
[14] |
Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 |
[15] |
Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973 |
[16] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[17] |
Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209 |
[18] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[19] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[20] |
Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623 |
2019 Impact Factor: 1.311
Tools
Metrics
Other articles
by authors
[Back to Top]