American Institute of Mathematical Sciences

Advanced
2013, 10(3): 939-957. doi: 10.3934/mbe.2013.10.939

Computational modeling approaches to studying the dynamics of oncolytic viruses

Dominik Wodarz 1,

1. 

Department of Ecology and Evolutionary Biology, University of California, 321 Steinhaus Hall, Irvine, CA 92617, United States

Received  July 2012 Revised  February 2013 Published  April 2013

Oncolytic viruses specifically infect cancer cells, replicate in them, kill them, and spread to further tumor cells. They represent a targeted treatment approach that is promising in principle, but consistent success has yet to be observed. Mathematical models can play an important role in analyzing the dynamics between oncolytic viruses and a growing tumor cell population, providing insights that can be useful for the further development of this therapy approach. This article reviews different mathematical modeling approaches ranging from ordinary differential equations to spatially explicit agent-based models. Problems of model robustness are discussed and so are some clinically important insight derived from the models.
Keywords: therapy, mathematical models, Oncolytic viruses, spatial dynamics., cancer, population dynamics.
Mathematics Subject Classification: 92D2.
Citation: Dominik Wodarz. Computational modeling approaches to studying the dynamics of oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 939-957. doi: 10.3934/mbe.2013.10.939
References:
[1]

J. C. Bell, Oncolytic viruses: What's next?, Curr. Cancer Drug Targets, 7 (2007), 127-131. Google Scholar

[2]

J. C. Bell, B. Lichty and D. Stojdl, Getting oncolytic virus therapies off the ground, Cancer Cell, 4 (2003), 7-11. Google Scholar

[3]

A. M. Crompton and D. H. Kirn, From ONYX-015 to armed vaccinia viruses: The education and evolution of oncolytic virus development, Curr. Cancer Drug Targets, 7 (2007), 133-139. Google Scholar

[4]

J. J. Davis and B. Fang, Oncolytic virotherapy for cancer treatment: Challenges and solutions, J. Gene. Med., 7 (2005), 1380-1389. Google Scholar

[5]

J. M. Kaplan, Adenovirus-based cancer gene therapy, Curr. Gene Ther., 5 (2005), 595-605. Google Scholar

[6]

E. Kelly and S. J. Russell, History of oncolytic viruses: Genesis to genetic engineering, Mol. Ther., 15 (2007), 651-659. Google Scholar

[7]

D. H. Kirn and F. McCormick, Replicating viruses as selective cancer therapeutics, Mol. Med. Today, 2 (1996), 519-527. Google Scholar

[8]

F. McCormick, Cancer-specific viruses and the development of ONYX-015, Cancer Biol. Ther., 2 (2003), S157-60. Google Scholar

[9]

F. McCormick, Future prospects for oncolytic therapy, Oncogene, 24 (2005), 7817-7819. Google Scholar

[10]

C. C. O'Shea, Viruses - seeking and destroying the tumor program, Oncogene, 24 (2005), 7640-7655. Google Scholar

[11]

K. A. Parato, et. al., Recent progress in the battle between oncolytic viruses and tumours, Nat. Rev. Cancer, 5 (2005), 965-976. Google Scholar

[12]

D. E. Post, et. al., Cancer scene investigation: how a cold virus became a tumor killer, Future Oncol., 1 (2005), 247-258. Google Scholar

[13]

M. S. Roberts, et. al., Naturally oncolytic viruses, Curr. Opin. Mol. Ther., 8 (2006), 314-321. doi: 10.1080/08898480600950473.  Google Scholar

[14]

M. J. Vaha-Koskela, J. E. Heikkila and A. E. Hinkkanen, Oncolytic viruses in cancer therapy, Cancer Lett., (2007). Google Scholar

[15]

H. H. Wong, N. R. Lemoine and Y. Wang, Oncolytic viruses for cancer therapy: Overcoming the obstacles, Viruses, 2 (2010), 78-106. Google Scholar

[16]

D. Koppers-Lalic and R. C. Hoeben, Non-human viruses developed as therapeutic agent for use in humans, Rev. Med. Virol, 21 (2011), 227-239. Google Scholar

[17]

R. L. Martuza, et. al., Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856. Google Scholar

[18]

K. Garber, China approves world's first oncolytic virus therapy for cancer treatment, J. Natl. Cancer Inst., 98 (2006), 298-300. Google Scholar

[19]

R. M. Eager and J. Nemunaitis, Clinical development directions in oncolytic viral therapy, Cancer Gene. Ther., 18 (2011), 305-317. Google Scholar

[20]

D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Cancer Res., 61 (2001), 3501-3507. Google Scholar

[21]

D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Hum. Gene. Ther., 14 (2003), 153-159. Google Scholar

[22]

Z. Bajzer, et. al., Modeling of cancer virotherapy with recombinant measles viruses, J. Theor. Biol., 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016.  Google Scholar

[23]

M. Biesecker, et. al., Optimization of virotherapy for cancer, Bull. Math. Biol., 72 (2010), 469-489. doi: 10.1007/s11538-009-9456-0.  Google Scholar

[24]

D. Dingli, et. al., Mathematical modeling of cancer radiovirotherapy, Math. Biosci., 199 (2006), 55-78. doi: 10.1016/j.mbs.2005.11.001.  Google Scholar

[25]

D. Dingli, et. al., Dynamics of multiple myeloma tumor therapy with a recombinant measles virus, Cancer Gene Ther., 16 (2009), 873-882. Google Scholar

[26]

A. Friedman, et. al., Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Cancer Res., 66 (2006), 2314-2319. Google Scholar

[27]

G. P. Karev, A. S. Novozhilov and E. V. Koonin, Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics, Biol. Direct, 1 (2006), pp. 30. Google Scholar

[28]

N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, J. Theor. Biol., 263 (2010), 530-543. doi: 10.1016/j.jtbi.2010.01.009.  Google Scholar

[29]

A. S. Novozhilov, et. al., Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models, Biol. Direct, 1 (2006), pp. 6. Google Scholar

[30]

L. M. Wein, J. T. Wu and D. H. Kirn, Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: Implications for virus design and delivery, Cancer Res., 63 (2003), 1317-1324. Google Scholar

[31]

D. Wodarz, Computational approaches to study oncolytic virus therapy: Insights and challenges, Gene Therapy and Molecular Biology, 8 (2004), 137-146. Google Scholar

[32]

D. Wodarz, Use of oncolytic viruses for the eradication of drug-resistant cancer cells, J. R. Soc. Interface, 6 (2009), 179-186. Google Scholar

[33]

D. Wodarz and N. Komarova, Towards predictive computational models of oncolytic virus therapy: Basis for experimental validation and model selection, PLoS ONE, 4 (2009), e4271. Google Scholar

[34]

N. Bagheri, et. al., A dynamical systems model for combinatorial cancer therapy enhances oncolytic adenovirus efficacy by MEK-inhibition, PLoS Comput. Biol., 7 (2011), e1001085. Google Scholar

[35]

R. Zurakowski and D. Wodarz, Model-driven approaches for in vitro combination therapy using ONYX-015 replicating oncolytic adenovirus, J. Theor. Biol., 245 (2007), 1-8. doi: 10.1016/j.jtbi.2006.09.029.  Google Scholar

[36]

W. Mok, et. al., Mathematical modeling of herpes simplex virus distribution in solid tumors: Implications for cancer gene therapy, Clin. Cancer Res., 15 (2009), 2352-2360. Google Scholar

[37]

L. R. Paiva, et. al., A multiscale mathematical model for oncolytic virotherapy, Cancer Res., 69 (2009), 1205-1211. Google Scholar

[38]

C. L. Reis, et. al., In silico evolutionary dynamics of tumour virotherapy, Integr. Biol. (Camb), 2 (2010), 41-45. Google Scholar

[39]

L. You, C. T. Yang and D. M. Jablons, ONYX-015 works synergistically with chemotherapy in lung cancer cell lines and primary cultures freshly made from lung cancer patients, Cancer Res., 60 (2000), 1009-1013. Google Scholar

[40]

A. Chahlavi, et. al., Replication-competent herpes simplex virus vector G207 and cisplatin combination therapy for head and neck squamous cell carcinoma, Neoplasia, 1 (1999), 162-169. Google Scholar

[41]

I. A. Rodriguez-Brenes, N. L. Komarova and D. Wodarz, Evolutionary dynamics of feedback escape and the development of stem-cell-driven cancers, Proc. Natl. Acad. Sci. U S A, 108 (2011), 18983-18988. Google Scholar

[42]

D. Wodarz, et. al., Complex spatial dynamics of oncolytic viruses in vitro: Mathematical and experimental approaches, PLoS Comput. Biol., 8 (2012), e1002547. Google Scholar

[43]

K. Sato, H. Matsuda and A. Sasaki, Pathogen invasion and host extinction in lattice structured populations, Journal of Mathematical Biology, 32 (1994), 251-268. Google Scholar

[44]

A. M. Deroos, E. Mccauley and W. G. Wilson, Mobility versus density-limited predator prey dynamics on different spatial scales, Proceedings of the Royal Society of London Series B-Biological Sciences, 246 (1991), 117-122. Google Scholar

[45]

M. Pascual, P. Mazzega and S. A. Levin, Oscillatory dynamics and spatial scale: The role of noise and unresolved pattern, Ecology, 82 (2001), 2357-2369. Google Scholar

[46]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", 1991, ().   Google Scholar

[47]

M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology,", 2000: Oxford University Press., ().   Google Scholar

[48]

M. P. Hassell, "The Spatial and Temporal Dynamics of Host-Parasitoid Interactions,", 2000, ().   Google Scholar

show all references

References:
[1]

J. C. Bell, Oncolytic viruses: What's next?, Curr. Cancer Drug Targets, 7 (2007), 127-131. Google Scholar

[2]

J. C. Bell, B. Lichty and D. Stojdl, Getting oncolytic virus therapies off the ground, Cancer Cell, 4 (2003), 7-11. Google Scholar

[3]

A. M. Crompton and D. H. Kirn, From ONYX-015 to armed vaccinia viruses: The education and evolution of oncolytic virus development, Curr. Cancer Drug Targets, 7 (2007), 133-139. Google Scholar

[4]

J. J. Davis and B. Fang, Oncolytic virotherapy for cancer treatment: Challenges and solutions, J. Gene. Med., 7 (2005), 1380-1389. Google Scholar

[5]

J. M. Kaplan, Adenovirus-based cancer gene therapy, Curr. Gene Ther., 5 (2005), 595-605. Google Scholar

[6]

E. Kelly and S. J. Russell, History of oncolytic viruses: Genesis to genetic engineering, Mol. Ther., 15 (2007), 651-659. Google Scholar

[7]

D. H. Kirn and F. McCormick, Replicating viruses as selective cancer therapeutics, Mol. Med. Today, 2 (1996), 519-527. Google Scholar

[8]

F. McCormick, Cancer-specific viruses and the development of ONYX-015, Cancer Biol. Ther., 2 (2003), S157-60. Google Scholar

[9]

F. McCormick, Future prospects for oncolytic therapy, Oncogene, 24 (2005), 7817-7819. Google Scholar

[10]

C. C. O'Shea, Viruses - seeking and destroying the tumor program, Oncogene, 24 (2005), 7640-7655. Google Scholar

[11]

K. A. Parato, et. al., Recent progress in the battle between oncolytic viruses and tumours, Nat. Rev. Cancer, 5 (2005), 965-976. Google Scholar

[12]

D. E. Post, et. al., Cancer scene investigation: how a cold virus became a tumor killer, Future Oncol., 1 (2005), 247-258. Google Scholar

[13]

M. S. Roberts, et. al., Naturally oncolytic viruses, Curr. Opin. Mol. Ther., 8 (2006), 314-321. doi: 10.1080/08898480600950473.  Google Scholar

[14]

M. J. Vaha-Koskela, J. E. Heikkila and A. E. Hinkkanen, Oncolytic viruses in cancer therapy, Cancer Lett., (2007). Google Scholar

[15]

H. H. Wong, N. R. Lemoine and Y. Wang, Oncolytic viruses for cancer therapy: Overcoming the obstacles, Viruses, 2 (2010), 78-106. Google Scholar

[16]

D. Koppers-Lalic and R. C. Hoeben, Non-human viruses developed as therapeutic agent for use in humans, Rev. Med. Virol, 21 (2011), 227-239. Google Scholar

[17]

R. L. Martuza, et. al., Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856. Google Scholar

[18]

K. Garber, China approves world's first oncolytic virus therapy for cancer treatment, J. Natl. Cancer Inst., 98 (2006), 298-300. Google Scholar

[19]

R. M. Eager and J. Nemunaitis, Clinical development directions in oncolytic viral therapy, Cancer Gene. Ther., 18 (2011), 305-317. Google Scholar

[20]

D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Cancer Res., 61 (2001), 3501-3507. Google Scholar

[21]

D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Hum. Gene. Ther., 14 (2003), 153-159. Google Scholar

[22]

Z. Bajzer, et. al., Modeling of cancer virotherapy with recombinant measles viruses, J. Theor. Biol., 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016.  Google Scholar

[23]

M. Biesecker, et. al., Optimization of virotherapy for cancer, Bull. Math. Biol., 72 (2010), 469-489. doi: 10.1007/s11538-009-9456-0.  Google Scholar

[24]

D. Dingli, et. al., Mathematical modeling of cancer radiovirotherapy, Math. Biosci., 199 (2006), 55-78. doi: 10.1016/j.mbs.2005.11.001.  Google Scholar

[25]

D. Dingli, et. al., Dynamics of multiple myeloma tumor therapy with a recombinant measles virus, Cancer Gene Ther., 16 (2009), 873-882. Google Scholar

[26]

A. Friedman, et. al., Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Cancer Res., 66 (2006), 2314-2319. Google Scholar

[27]

G. P. Karev, A. S. Novozhilov and E. V. Koonin, Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics, Biol. Direct, 1 (2006), pp. 30. Google Scholar

[28]

N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, J. Theor. Biol., 263 (2010), 530-543. doi: 10.1016/j.jtbi.2010.01.009.  Google Scholar

[29]

A. S. Novozhilov, et. al., Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models, Biol. Direct, 1 (2006), pp. 6. Google Scholar

[30]

L. M. Wein, J. T. Wu and D. H. Kirn, Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: Implications for virus design and delivery, Cancer Res., 63 (2003), 1317-1324. Google Scholar

[31]

D. Wodarz, Computational approaches to study oncolytic virus therapy: Insights and challenges, Gene Therapy and Molecular Biology, 8 (2004), 137-146. Google Scholar

[32]

D. Wodarz, Use of oncolytic viruses for the eradication of drug-resistant cancer cells, J. R. Soc. Interface, 6 (2009), 179-186. Google Scholar

[33]

D. Wodarz and N. Komarova, Towards predictive computational models of oncolytic virus therapy: Basis for experimental validation and model selection, PLoS ONE, 4 (2009), e4271. Google Scholar

[34]

N. Bagheri, et. al., A dynamical systems model for combinatorial cancer therapy enhances oncolytic adenovirus efficacy by MEK-inhibition, PLoS Comput. Biol., 7 (2011), e1001085. Google Scholar

[35]

R. Zurakowski and D. Wodarz, Model-driven approaches for in vitro combination therapy using ONYX-015 replicating oncolytic adenovirus, J. Theor. Biol., 245 (2007), 1-8. doi: 10.1016/j.jtbi.2006.09.029.  Google Scholar

[36]

W. Mok, et. al., Mathematical modeling of herpes simplex virus distribution in solid tumors: Implications for cancer gene therapy, Clin. Cancer Res., 15 (2009), 2352-2360. Google Scholar

[37]

L. R. Paiva, et. al., A multiscale mathematical model for oncolytic virotherapy, Cancer Res., 69 (2009), 1205-1211. Google Scholar

[38]

C. L. Reis, et. al., In silico evolutionary dynamics of tumour virotherapy, Integr. Biol. (Camb), 2 (2010), 41-45. Google Scholar

[39]

L. You, C. T. Yang and D. M. Jablons, ONYX-015 works synergistically with chemotherapy in lung cancer cell lines and primary cultures freshly made from lung cancer patients, Cancer Res., 60 (2000), 1009-1013. Google Scholar

[40]

A. Chahlavi, et. al., Replication-competent herpes simplex virus vector G207 and cisplatin combination therapy for head and neck squamous cell carcinoma, Neoplasia, 1 (1999), 162-169. Google Scholar

[41]

I. A. Rodriguez-Brenes, N. L. Komarova and D. Wodarz, Evolutionary dynamics of feedback escape and the development of stem-cell-driven cancers, Proc. Natl. Acad. Sci. U S A, 108 (2011), 18983-18988. Google Scholar

[42]

D. Wodarz, et. al., Complex spatial dynamics of oncolytic viruses in vitro: Mathematical and experimental approaches, PLoS Comput. Biol., 8 (2012), e1002547. Google Scholar

[43]

K. Sato, H. Matsuda and A. Sasaki, Pathogen invasion and host extinction in lattice structured populations, Journal of Mathematical Biology, 32 (1994), 251-268. Google Scholar

[44]

A. M. Deroos, E. Mccauley and W. G. Wilson, Mobility versus density-limited predator prey dynamics on different spatial scales, Proceedings of the Royal Society of London Series B-Biological Sciences, 246 (1991), 117-122. Google Scholar

[45]

M. Pascual, P. Mazzega and S. A. Levin, Oscillatory dynamics and spatial scale: The role of noise and unresolved pattern, Ecology, 82 (2001), 2357-2369. Google Scholar

[46]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", 1991, ().   Google Scholar

[47]

M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology,", 2000: Oxford University Press., ().   Google Scholar

[48]

M. P. Hassell, "The Spatial and Temporal Dynamics of Host-Parasitoid Interactions,", 2000, ().   Google Scholar

[1]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565
[2]

Alexander S. Bratus, Svetlana Yu. Kovalenko, Elena Fimmel. On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells. Mathematical Biosciences & Engineering, 2015, 12 (1) : 163-183. doi: 10.3934/mbe.2015.12.163
[3]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[4]

Sophia R-J Jang, Hsiu-Chuan Wei. On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021184
[5]

Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591
[6]

Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021140
[7]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3419-3440. doi: 10.3934/dcdss.2020426
[8]

Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623
[9]

B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure & Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19
[10]

Robert Carlson. Myopic models of population dynamics on infinite networks. Networks & Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477
[11]

Peixuan Weng, Xiao-Qiang Zhao. Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 343-366. doi: 10.3934/dcds.2011.29.343
[12]

José Antonio Carrillo, Seung-Yeal Ha, Lorenzo Pareschi, Benedetto Piccoli. Special issue on mathematical models for collective dynamics. Networks & Heterogeneous Media, 2020, 15 (3) : i-i. doi: 10.3934/nhm.2020020
[13]

Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221
[14]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237
[15]

Avner Friedman. A hierarchy of cancer models and their mathematical challenges. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 147-159. doi: 10.3934/dcdsb.2004.4.147
[16]

Karen R. Ríos-Soto, Baojun Song, Carlos Castillo-Chavez. Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 199-222. doi: 10.3934/mbe.2011.8.199
[17]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761
[18]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051
[19]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[20]

Cecilia Cavaterra, Maurizio Grasselli. Asymptotic behavior of population dynamics models with nonlocal distributed delays. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 861-883. doi: 10.3934/dcds.2008.22.861

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences