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Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation
Non-Gaussian dynamics of a tumor growth system with immunization
1. | Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China, China |
2. | Institute for Pure and Applied Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States |
3. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 |
References:
[1] |
J. A. Adam, The dynamics of growth-factor-modified immune response to cancer growth: One dimensional models,, Mathl. Comput. Modelling, 17 (1993), 83.
doi: 10.1016/0895-7177(93)90041-V. |
[2] |
S. Albeverrio, B. Rüdiger and J. L. Wu, Invariant measures and symmetry property of lévy type operators,, Potential Analysis, 13 (2000), 147.
doi: 10.1023/A:1008705820024. |
[3] |
D. Applebaum, "Lévy Processes and Stochastic Calculus,", Cambridge Studies in Advanced Mathematics, (2004).
doi: 10.1017/CBO9780511755323. |
[4] |
F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra and G. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus brownian strategies,, Phys. Rev. Lett., 88 (2002).
doi: 10.1103/PhysRevLett.88.097901. |
[5] |
T. Bose and S. Trimper, Stochastic model for tumor growth with immunization,, Phys. Rev. E, 79 (2009).
doi: 10.1103/PhysRevE.79.051903. |
[6] |
J. R. Brannan, J. Duan and V. J. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics,, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, 133 (1999), 23.
doi: 10.1016/S0167-2789(99)00096-2. |
[7] |
H. Chen, J. Duan, X. Li and C. Zhang, A computational analysis for mean exit time under non-Gaussian lévy noises,, Applied Mathematics and Computation, 218 (2011), 1845.
doi: 10.1016/j.amc.2011.06.068. |
[8] |
Z. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet fractional laplacian,, J. European Math. Soc., 12 (2010), 1307.
doi: 10.4171/JEMS/231. |
[9] |
L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations,, J. Theoret. Biol., 238 (2006), 841.
doi: 10.1016/j.jtbi.2005.06.037. |
[10] |
J. R. R. Duarte, M. V. D. Vermelho and M. L. Lyra, Stochastic resonance of a periodically driven neuron under non-Gaussian noise,, Physica A, 387 (2008), 1446.
doi: 10.1016/j.physa.2007.11.011. |
[11] |
A. Fiasconaro, A. Ochab-Marcinek, B. Spagnolo and E. Gudowska-Nowak, Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment,, Eur. Phys. J. B, 65 (2008), 435.
doi: 10.1140/epjb/e2008-00246-2. |
[12] |
A. Fiasconaro and B. Spagnolo, Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response,, Phys. Rev. E, 74 (2006).
doi: 10.1103/PhysRevE.74.041904. |
[13] |
T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by lévy noise, preprint,, , (). Google Scholar |
[14] |
R. P. Garay and R. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions,, J. Theor. Biol., 73 (1978), 417.
doi: 10.1016/0022-5193(78)90150-9. |
[15] |
W. Horsthemke and R. Lefever, "Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology,", Springer Series in Synergetics, (1984).
|
[16] |
N. E. Humphries et al., Environmental context explains lévy and brownian movement patterns of marine predators,, Nature, 465 (2010), 1066.
doi: 10.1038/nature09116. |
[17] |
L. Jiang, X. Luo, D. Wu and S. Zhu, Stochastic properties of tumor growth driven by white lévy noise,, Modern Physics Letters B, 26 (2012).
doi: 10.1142/S0217984912501497. |
[18] |
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235.
doi: 10.1007/s002850050127. |
[19] |
A. E. Kyprianou, "Introductory Lectures on Fluctuations of Lévy Processes with Applications,", Springer-Verlag, (2006).
|
[20] |
R. Lefever and W. Horsthemk, Bistability in fluctuating environments. Implications in tumor immumology,, Bulletin of Mathematical Biology, 41 (1979), 469.
doi: 10.1007/BF02458325. |
[21] |
D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment,, Physics Letters A, 375 (2011), 886.
doi: 10.1016/j.physleta.2010.12.066. |
[22] |
M. Liao, The dirichlet problem of a discontinuous markov process,, A Chinese summary appears in Acta Math., 33 (1989), 9.
doi: 10.1007/BF02107618. |
[23] |
T. Naeh, M. M. Klosek, B. J. Matkowsky and Z. Schuss, A direct approach to the exit problem,, SIAM J. Appl. Math., 50 (1990), 595.
doi: 10.1137/0150036. |
[24] |
A. Ochab-Marcinek and E. Gudowska-Nowak, Population growth and control in stochastic models of cancer development,, Physica A, 343 (2004), 557.
doi: 10.1016/j.physa.2004.06.071. |
[25] |
I. Prigogine and R. Lefever, Stability problems in cancer growth and nucleation,, Comp. Biochem. Physiol, 67 (1980), 389.
doi: 10.1016/0305-0491(80)90326-0. |
[26] |
H. Qiao, X. Kan and J. Duan, Escape probability for stochastic dynamical systems with jumps,, Malliavin Calculus and Stochastic Analysis, 34 (2013), 195.
doi: 10.1007/978-1-4614-5906-4_9. |
[27] |
K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, (1990).
|
[28] |
D. Schertzer, M. Larchevêque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian lévy stable noises,, J. Math. Phys., 42 (2001), 200.
doi: 10.1063/1.1318734. |
[29] |
Z. Schuss, "Theory and Applications of Stochastic Differential Equations,", Wiley Series in Probability and Statistics, (1980).
|
[30] |
C. Zeng, X. Zhou and S. Tao, Cross-correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross-correlated noises,, J. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/49/495002. |
[31] |
C. Zeng and H. Wang, Colored noise enhanced stability in a tumor cell growth system under immune response,, J. Stat. Phys., 141 (2010), 889.
doi: 10.1007/s10955-010-0068-8. |
[32] |
C. Zeng, Effects of correlated noise in a tumor cell growth model in the presence of immune response,, Phys. Scr., 81 (2010).
doi: 10.1088/0031-8949/81/02/025009. |
[33] |
W. Zhong, Y. Shao and Z. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability,, Phys. Rev. E, 73 (2006).
doi: 10.1103/PhysRevE.73.060902. |
[34] |
W. Zhong, Y. Shao and Z. He, Spatiotemporal fluctuation-induced transition in a tumor model with immune surveillance,, Phys. Rev. E, 74 (2006).
doi: 10.1103/PhysRevE.74.011916. |
show all references
References:
[1] |
J. A. Adam, The dynamics of growth-factor-modified immune response to cancer growth: One dimensional models,, Mathl. Comput. Modelling, 17 (1993), 83.
doi: 10.1016/0895-7177(93)90041-V. |
[2] |
S. Albeverrio, B. Rüdiger and J. L. Wu, Invariant measures and symmetry property of lévy type operators,, Potential Analysis, 13 (2000), 147.
doi: 10.1023/A:1008705820024. |
[3] |
D. Applebaum, "Lévy Processes and Stochastic Calculus,", Cambridge Studies in Advanced Mathematics, (2004).
doi: 10.1017/CBO9780511755323. |
[4] |
F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra and G. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus brownian strategies,, Phys. Rev. Lett., 88 (2002).
doi: 10.1103/PhysRevLett.88.097901. |
[5] |
T. Bose and S. Trimper, Stochastic model for tumor growth with immunization,, Phys. Rev. E, 79 (2009).
doi: 10.1103/PhysRevE.79.051903. |
[6] |
J. R. Brannan, J. Duan and V. J. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics,, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, 133 (1999), 23.
doi: 10.1016/S0167-2789(99)00096-2. |
[7] |
H. Chen, J. Duan, X. Li and C. Zhang, A computational analysis for mean exit time under non-Gaussian lévy noises,, Applied Mathematics and Computation, 218 (2011), 1845.
doi: 10.1016/j.amc.2011.06.068. |
[8] |
Z. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet fractional laplacian,, J. European Math. Soc., 12 (2010), 1307.
doi: 10.4171/JEMS/231. |
[9] |
L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations,, J. Theoret. Biol., 238 (2006), 841.
doi: 10.1016/j.jtbi.2005.06.037. |
[10] |
J. R. R. Duarte, M. V. D. Vermelho and M. L. Lyra, Stochastic resonance of a periodically driven neuron under non-Gaussian noise,, Physica A, 387 (2008), 1446.
doi: 10.1016/j.physa.2007.11.011. |
[11] |
A. Fiasconaro, A. Ochab-Marcinek, B. Spagnolo and E. Gudowska-Nowak, Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment,, Eur. Phys. J. B, 65 (2008), 435.
doi: 10.1140/epjb/e2008-00246-2. |
[12] |
A. Fiasconaro and B. Spagnolo, Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response,, Phys. Rev. E, 74 (2006).
doi: 10.1103/PhysRevE.74.041904. |
[13] |
T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by lévy noise, preprint,, , (). Google Scholar |
[14] |
R. P. Garay and R. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions,, J. Theor. Biol., 73 (1978), 417.
doi: 10.1016/0022-5193(78)90150-9. |
[15] |
W. Horsthemke and R. Lefever, "Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology,", Springer Series in Synergetics, (1984).
|
[16] |
N. E. Humphries et al., Environmental context explains lévy and brownian movement patterns of marine predators,, Nature, 465 (2010), 1066.
doi: 10.1038/nature09116. |
[17] |
L. Jiang, X. Luo, D. Wu and S. Zhu, Stochastic properties of tumor growth driven by white lévy noise,, Modern Physics Letters B, 26 (2012).
doi: 10.1142/S0217984912501497. |
[18] |
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235.
doi: 10.1007/s002850050127. |
[19] |
A. E. Kyprianou, "Introductory Lectures on Fluctuations of Lévy Processes with Applications,", Springer-Verlag, (2006).
|
[20] |
R. Lefever and W. Horsthemk, Bistability in fluctuating environments. Implications in tumor immumology,, Bulletin of Mathematical Biology, 41 (1979), 469.
doi: 10.1007/BF02458325. |
[21] |
D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment,, Physics Letters A, 375 (2011), 886.
doi: 10.1016/j.physleta.2010.12.066. |
[22] |
M. Liao, The dirichlet problem of a discontinuous markov process,, A Chinese summary appears in Acta Math., 33 (1989), 9.
doi: 10.1007/BF02107618. |
[23] |
T. Naeh, M. M. Klosek, B. J. Matkowsky and Z. Schuss, A direct approach to the exit problem,, SIAM J. Appl. Math., 50 (1990), 595.
doi: 10.1137/0150036. |
[24] |
A. Ochab-Marcinek and E. Gudowska-Nowak, Population growth and control in stochastic models of cancer development,, Physica A, 343 (2004), 557.
doi: 10.1016/j.physa.2004.06.071. |
[25] |
I. Prigogine and R. Lefever, Stability problems in cancer growth and nucleation,, Comp. Biochem. Physiol, 67 (1980), 389.
doi: 10.1016/0305-0491(80)90326-0. |
[26] |
H. Qiao, X. Kan and J. Duan, Escape probability for stochastic dynamical systems with jumps,, Malliavin Calculus and Stochastic Analysis, 34 (2013), 195.
doi: 10.1007/978-1-4614-5906-4_9. |
[27] |
K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, (1990).
|
[28] |
D. Schertzer, M. Larchevêque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian lévy stable noises,, J. Math. Phys., 42 (2001), 200.
doi: 10.1063/1.1318734. |
[29] |
Z. Schuss, "Theory and Applications of Stochastic Differential Equations,", Wiley Series in Probability and Statistics, (1980).
|
[30] |
C. Zeng, X. Zhou and S. Tao, Cross-correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross-correlated noises,, J. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/49/495002. |
[31] |
C. Zeng and H. Wang, Colored noise enhanced stability in a tumor cell growth system under immune response,, J. Stat. Phys., 141 (2010), 889.
doi: 10.1007/s10955-010-0068-8. |
[32] |
C. Zeng, Effects of correlated noise in a tumor cell growth model in the presence of immune response,, Phys. Scr., 81 (2010).
doi: 10.1088/0031-8949/81/02/025009. |
[33] |
W. Zhong, Y. Shao and Z. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability,, Phys. Rev. E, 73 (2006).
doi: 10.1103/PhysRevE.73.060902. |
[34] |
W. Zhong, Y. Shao and Z. He, Spatiotemporal fluctuation-induced transition in a tumor model with immune surveillance,, Phys. Rev. E, 74 (2006).
doi: 10.1103/PhysRevE.74.011916. |
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