American Institute of Mathematical Sciences

Advanced
July  2021, 26(7): 3717-3735. doi: 10.3934/dcdsb.2020254

Analysis of non-Markovian effects in generalized birth-death models

Zhenquan Zhang , Meiling Chen , Jiajun Zhang and  Tianshou Zhou ,

Key Laboratory of Computational Mathematics, Guangdong Province and School of, Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

* Corresponding author: Tianshou Zhou

Received  October 2019 Revised  June 2020 Published  July 2021 Early access  August 2020

Birth-death processes are a fundamental reaction module for which we can find its prototypes in many scientific fields. For such a kind of module, if all the reaction events are Markovian, the reaction kinetics is simple. However, experimentally observable quantities are in general consequences of a series of reactions, implying that the synthesis of a macromolecule in general involve multiple middle reaction steps with some reactions that would not be specified by experiments. This multistep process can create molecular memory between reaction events, leading to non-Markovian behavior. Based on the theoretical framework established in a recent paper published in [39], we find that the effect of non-Markovianity is equivalent to the introduction of a feedback, non-Markovianity always amplifies the mean level of the product if the death reaction is non-Markovian but always reduces the mean level if the birth reaction is non-Markovian, and in contrast to Markovianity, non-Markovianity can reduce or amplify the product noise, depending on the details of waiting-time distributions characterizing reaction events. Examples analysis indicates that non-Markovianity, whose effects were neglected in previous studies, can significantly impact gene expression.

Keywords: Birth-death process, non-Markovianity, waiting-time distribution.
Mathematics Subject Classification: Primary: 60G07, 60J27; Secondary: 92B05.
Citation: Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
References:
[1]

T. Aquino and M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett., 119 (2017), 230601. doi: 10.1103/PhysRevLett.119.230601.

[2]

A. L. Barabasi, The origin of bursts and heavy tails in human dynamics, Nature, 435 (2005), 207-211.  doi: 10.1038/nature03459.

[3]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1 discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117.

[4]

D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 14593-14598.  doi: 10.1073/pnas.0503858102.

[5]

T. Brett and T. Galla, Stochastic processes with distributed delays: Chemical Langevin equation and linear noise approximation, Phys. Rev. Lett., 110 (2013), 250601. doi: 10.1103/PhysRevLett.110.250601.

[6]

A. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92 (2004), 108501. doi: 10.1103/PhysRevLett.92.108501.

[7]

I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Modern Phys., 89 (2017), 015001. doi: 10.1103/RevModPhys.89.015001.

[8]

J. Delvenne, R. Lambiotte and L. E. C. Rocha, Diffusion on networked systems is a question of time or structure, Nat. Commun., 6 (2015), 1-10.  doi: 10.1038/ncomms8366.

[9]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley & Sons, New York, 2008.

[10]

C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer-Verlag, Berlin, 2009.

[11]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.

[12]

J. P. Gleeson, K. P. OŚullivan, R. A. Baños and Y. Moreno, Effects of network structure, competition and memory time on social spreading phenomena, Phys. Rev. X, 6 (2016), 021019. doi: 10.1103/PhysRevX.6.021019.

[13]

T. Guérin, O. Bénichou and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem., 4 (2012), 568-573. 

[14]

C. V. Harper, B. Finkenstädt, D. J. Woodcock, S. Friedrichsen, S. Semprini, L. Ashall, et al., Dynamic analysis of stochastic transcription cycles, PLoS Biol., 9 (2011), e1000607. doi: 10.1371/journal.pbio.1000607.

[15]

H. W. Hethcote and P. v. d. Driessche, A SIS epidemic model with variable population size and a delay, J. Math. Biol., 34 (1995), 177-194.  doi: 10.1007/BF00178772.

[16]

J. C. Jaeger and G. Newstead, An Introduction to the Laplace Transformation with Engineering Applications, Methuen & Co., Ltd., London, John Wiley & Sons, Inc., New York, NY, 1949.

[17]

T. Jia and R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory, Phys. Rev. Lett., 106 (2011), 058102. doi: 10.1103/PhysRevLett.106.058102.

[18]

H. H. Jo, J. I. Perotti, K. Kaski and J. Kertész, Analytically solvable model of spreading dynamics with non-Poissonian processes, Phys. Rev. X, 4 (2014), 011041. doi: 10.1103/PhysRevX.4.011041.

[19]

I. Z. Kiss, G. Röst and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks, Phys. Rev. Lett., 115 (2015), 078701. doi: 10.1103/PhysRevLett.115.078701.

[20]

N. Kumar, A. Singh and R. V. Kulkarni, Transcriptional bursting in gene expression: Analytical results for general stochastic models, PLoS Comput. Biol., 11 (2015), e1004292. doi: 10.1371/journal.pcbi.1004292.

[21]

D. R. Larson, What do expression dynamics tell us about the mechanism of transcription?, Curr. Opin. Genet. Dev., 21 (2011), 591-599.  doi: 10.1016/j.gde.2011.07.010.

[22]

N. Masuda, M. A. Porter and R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1-58.  doi: 10.1016/j.physrep.2017.07.007.

[23]

A. S. Novozhilov, G. P. Karev and E. V. Koonin, Biological applications of the theory of birth-and-death processes, Briefings in Bioinformatics, 7 (2006), 70-85.  doi: 10.1093/bib/bbk006.

[24]

E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, Vol. 796, John Wiley & Sons, New York, 2008. doi: 10.1002/9780470721872.

[25]

J. Peccoud and B. Ycart, Markovian modeling of gene product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.  doi: 10.1006/tpbi.1995.1027.

[26]

J. M. Pedraza and J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.  doi: 10.1126/science.1144331.

[27]

A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309.

[28]

M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman and J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025. 

[29]

I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C. J. Tessone and F. Schweitzer, Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks, Nat. Commun., 5 (2014), 1-9.  doi: 10.1038/ncomms6024.

[30]

A. Schwabe, K. N. Rybakova and F. J. Bruggeman, Transcription stochasticity of complex gene regulation models, Biophys. J., 103 (2012), 1152-1161.  doi: 10.1016/j.bpj.2012.07.011.

[31]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105.

[32]

M. Starnini, J. P. Gleeson and M. Boguñá, Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes, Phys. Rev. Lett., 118 (2017), 128301. doi: 10.1103/PhysRevLett.118.128301.

[33]

P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, T. E. Chan, M. P. Stumpf, C. P. Please, S. D. Howison, F. Arai and B. D. MacArthur, Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.  doi: 10.1016/j.cels.2017.08.009.

[34]

D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.

[35]

P. Thomas, N. Popović and R. Grima, Phenotypic switching in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 6994-6999.  doi: 10.1073/pnas.1400049111.

[36]

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007.

[37]

P. Van Mieghem and R. Van de Bovenkamp, Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks, Phys. Rev. Lett., 110 (2013), 108701. doi: 10.1103/PhysRevLett.110.108701.

[38]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.

[39]

J. J. Zhang and T. S. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U. S. A., 116 (2019), 23542-23550.  doi: 10.1073/pnas.1913926116.

[40]

J. J. Zhang, Q. Nie and T. S. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109. doi: 10.1063/1.4950767.

show all references

References:
[1]

T. Aquino and M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett., 119 (2017), 230601. doi: 10.1103/PhysRevLett.119.230601.

[2]

A. L. Barabasi, The origin of bursts and heavy tails in human dynamics, Nature, 435 (2005), 207-211.  doi: 10.1038/nature03459.

[3]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1 discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117.

[4]

D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 14593-14598.  doi: 10.1073/pnas.0503858102.

[5]

T. Brett and T. Galla, Stochastic processes with distributed delays: Chemical Langevin equation and linear noise approximation, Phys. Rev. Lett., 110 (2013), 250601. doi: 10.1103/PhysRevLett.110.250601.

[6]

A. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92 (2004), 108501. doi: 10.1103/PhysRevLett.92.108501.

[7]

I. De Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Modern Phys., 89 (2017), 015001. doi: 10.1103/RevModPhys.89.015001.

[8]

J. Delvenne, R. Lambiotte and L. E. C. Rocha, Diffusion on networked systems is a question of time or structure, Nat. Commun., 6 (2015), 1-10.  doi: 10.1038/ncomms8366.

[9]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley & Sons, New York, 2008.

[10]

C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer-Verlag, Berlin, 2009.

[11]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.

[12]

J. P. Gleeson, K. P. OŚullivan, R. A. Baños and Y. Moreno, Effects of network structure, competition and memory time on social spreading phenomena, Phys. Rev. X, 6 (2016), 021019. doi: 10.1103/PhysRevX.6.021019.

[13]

T. Guérin, O. Bénichou and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem., 4 (2012), 568-573. 

[14]

C. V. Harper, B. Finkenstädt, D. J. Woodcock, S. Friedrichsen, S. Semprini, L. Ashall, et al., Dynamic analysis of stochastic transcription cycles, PLoS Biol., 9 (2011), e1000607. doi: 10.1371/journal.pbio.1000607.

[15]

H. W. Hethcote and P. v. d. Driessche, A SIS epidemic model with variable population size and a delay, J. Math. Biol., 34 (1995), 177-194.  doi: 10.1007/BF00178772.

[16]

J. C. Jaeger and G. Newstead, An Introduction to the Laplace Transformation with Engineering Applications, Methuen & Co., Ltd., London, John Wiley & Sons, Inc., New York, NY, 1949.

[17]

T. Jia and R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory, Phys. Rev. Lett., 106 (2011), 058102. doi: 10.1103/PhysRevLett.106.058102.

[18]

H. H. Jo, J. I. Perotti, K. Kaski and J. Kertész, Analytically solvable model of spreading dynamics with non-Poissonian processes, Phys. Rev. X, 4 (2014), 011041. doi: 10.1103/PhysRevX.4.011041.

[19]

I. Z. Kiss, G. Röst and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks, Phys. Rev. Lett., 115 (2015), 078701. doi: 10.1103/PhysRevLett.115.078701.

[20]

N. Kumar, A. Singh and R. V. Kulkarni, Transcriptional bursting in gene expression: Analytical results for general stochastic models, PLoS Comput. Biol., 11 (2015), e1004292. doi: 10.1371/journal.pcbi.1004292.

[21]

D. R. Larson, What do expression dynamics tell us about the mechanism of transcription?, Curr. Opin. Genet. Dev., 21 (2011), 591-599.  doi: 10.1016/j.gde.2011.07.010.

[22]

N. Masuda, M. A. Porter and R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1-58.  doi: 10.1016/j.physrep.2017.07.007.

[23]

A. S. Novozhilov, G. P. Karev and E. V. Koonin, Biological applications of the theory of birth-and-death processes, Briefings in Bioinformatics, 7 (2006), 70-85.  doi: 10.1093/bib/bbk006.

[24]

E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, Vol. 796, John Wiley & Sons, New York, 2008. doi: 10.1002/9780470721872.

[25]

J. Peccoud and B. Ycart, Markovian modeling of gene product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.  doi: 10.1006/tpbi.1995.1027.

[26]

J. M. Pedraza and J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.  doi: 10.1126/science.1144331.

[27]

A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309.

[28]

M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman and J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025. 

[29]

I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C. J. Tessone and F. Schweitzer, Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks, Nat. Commun., 5 (2014), 1-9.  doi: 10.1038/ncomms6024.

[30]

A. Schwabe, K. N. Rybakova and F. J. Bruggeman, Transcription stochasticity of complex gene regulation models, Biophys. J., 103 (2012), 1152-1161.  doi: 10.1016/j.bpj.2012.07.011.

[31]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105.

[32]

M. Starnini, J. P. Gleeson and M. Boguñá, Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes, Phys. Rev. Lett., 118 (2017), 128301. doi: 10.1103/PhysRevLett.118.128301.

[33]

P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, T. E. Chan, M. P. Stumpf, C. P. Please, S. D. Howison, F. Arai and B. D. MacArthur, Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.  doi: 10.1016/j.cels.2017.08.009.

[34]

D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.

[35]

P. Thomas, N. Popović and R. Grima, Phenotypic switching in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 111 (2014), 6994-6999.  doi: 10.1073/pnas.1400049111.

[36]

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007.

[37]

P. Van Mieghem and R. Van de Bovenkamp, Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks, Phys. Rev. Lett., 110 (2013), 108701. doi: 10.1103/PhysRevLett.110.108701.

[38]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.

[39]

J. J. Zhang and T. S. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U. S. A., 116 (2019), 23542-23550.  doi: 10.1073/pnas.1913926116.

[40]

J. J. Zhang, Q. Nie and T. S. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109. doi: 10.1063/1.4950767.

Figure 1.  Schematic diagram for birth-death reaction process (A), where $ \psi_{1} \left(t\right) $ and $ \psi_{2} \left(t;n\right) $ are interreaction waiting-time distributions, which may be exponential (B) or non-exponential (C). Here $ n $ represents the number of molecules of reactive species $ X $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Difference between dynamic probability distributions obtained in the original non-Markovian model (curves with empty circles) and in the constructed Markovian model (colored curves), both being used in the modeling of a gene transcription process. Parameter values are set as $ k_{1} = 3,\lambda_{1} = 90,\lambda_{2} = 1 $, and the initial conditions are set as the same in two cases. In the diagram, '$ t = \inf $' means that the distributions after $ t>10 $ are approximately the same, implying that the stationary distribution exists
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Characteristics of stationary protein distribution in a generalized model of constitutive gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $ k_{d} = 1,\lambda_{d} = 1 $ and (C) $ k_{b} = 1,\lambda_{b} = 10 $. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Characteristics of stationary protein distribution in a generalized model of bursty gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $ \mu = 10,\delta = 1 $, $ b = 2,k_{on} = 1,\lambda_{on} = 0.5,k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{on} = 4,\lambda_{on} = 4 $ and the other parameter values are kept unchanged for non-Markovianity; (C)$ \mu = 10,\delta = 1,b = 2 $, $ k_{on} = 1,\lambda_{on} = 0.5 $, $ k_{off} = 1,\lambda_{off} = 0.1 $ for Markovianity, whereas $ k_{off} = 4,\lambda_{off} = 0.4 $ and the other parameter values are kept unchanged for non-Markovianity. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hilla Behar, Alexandra Agranovich, Yoram Louzoun. Diffusion rate determines balance between extinction and proliferation in birth-death processes. Mathematical Biosciences & Engineering, 2013, 10 (3) : 523-550. doi: 10.3934/mbe.2013.10.523
[2]

Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1415-1433. doi: 10.3934/jimo.2019009
[3]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial and Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593
[4]

Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations and Control Theory, 2022, 11 (3) : 895-923. doi: 10.3934/eect.2021030
[5]

Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67
[6]

Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005
[7]

Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529
[8]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735
[9]

Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107
[10]

Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
[11]

Abed Boulouz. A spatially and size-structured population model with unbounded birth process. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022038
[12]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
[13]

Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737
[14]

Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure and Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637
[15]

Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032
[16]

Andrey Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Conference Publications, 2007, 2007 (Special) : 938-945. doi: 10.3934/proc.2007.2007.938
[17]

Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021078
[18]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
[19]

Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153
[20]

Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic and Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (377)
  • HTML views (331)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy