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STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS

Year 2018, Volume: 19 Issue: 1, 50 - 57, 31.03.2018
https://doi.org/10.18038/aubtda.323014

Abstract

This work has been presented at the
”International Conference on Mathematics and Engineering, 10-12 May, 2017,
Istanbul, Turkey”. In this paper we introduce a stability and Hopf bifurcation
analysis of a reaction diffusion system which models the interaction between
endothelial cells and the inhibitor. Then, we investigate the stability of the
positive equilibrium solutions under some conditions. We also show the
existence of a Hopf bifurcation and provide some figures to show that the
equilibrium solutions are indeed asymptotically stable.

References

  • [1] Tang X, Song Y. Bifurcation analysis and Turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality. Chaos, Soli-tons and Fractals 2015;81: 303-314.
  • [2] Yang R, Song Y. Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model. Nonlinear Analysis: Real World Applications 2016;31:356-387.
  • [3] Dil˜ao R. Turing instabilities and patterns near a Hopf bifurcation. Applied Math-ematics and Computation 2005;164:391-414.
  • [4] Ling W, Hongyong Z. Hopf bifurcation and Turing instability of 2-D Lengyel-Epstein system with reaction-diffusion terms. Applied Mathematics and Computation 2013;219:9229-9244.
  • [5] Zhang JF, Li WT, Yan XP. Hopf bifurcation and Turing instability in spa-tial homogeneous and inhomogeneous predator-prey models. Applied Mathematics and Computation 2011;218:1883-1893.
  • [6] Song Y, Zhang T, Peng Y. Turing-Hopf bifurcation in the reaction-diffusion equations and its applications. Commun Nonlinear Sci Numer Simulat 2016;33:229-258.
  • [7] Pamuk S, G¨urb¨uz A. Stability analysis of the steady-state solution of a mathe-matical model in tumor angiogenesis. AIP Conference Proceedings 2004;729: 369 .
  • [8] Levine HA, Pamuk S, Sleeman BD, Hamilton MN. A mathematical model of capillary formation and development in tumour angiogenesis: penetration into the stroma. Bull. Math. Biol 2001;63:801-863.
  • [9] Pamuk S, Çay I. A 2D Mathematical Model for Tumor Angiogenesis: The Roles of Endothelials, Pericytes and Macrophages in the ECM. (in preperation).
  • [10] Karaoglu E, Merdan H. Hopf bifurcation of a ratio - dependent predator-prey model involving two discrete maturation time delays. Chaos, Solitons & Fractals 2014;68:159-168.
Year 2018, Volume: 19 Issue: 1, 50 - 57, 31.03.2018
https://doi.org/10.18038/aubtda.323014

Abstract

References

  • [1] Tang X, Song Y. Bifurcation analysis and Turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality. Chaos, Soli-tons and Fractals 2015;81: 303-314.
  • [2] Yang R, Song Y. Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model. Nonlinear Analysis: Real World Applications 2016;31:356-387.
  • [3] Dil˜ao R. Turing instabilities and patterns near a Hopf bifurcation. Applied Math-ematics and Computation 2005;164:391-414.
  • [4] Ling W, Hongyong Z. Hopf bifurcation and Turing instability of 2-D Lengyel-Epstein system with reaction-diffusion terms. Applied Mathematics and Computation 2013;219:9229-9244.
  • [5] Zhang JF, Li WT, Yan XP. Hopf bifurcation and Turing instability in spa-tial homogeneous and inhomogeneous predator-prey models. Applied Mathematics and Computation 2011;218:1883-1893.
  • [6] Song Y, Zhang T, Peng Y. Turing-Hopf bifurcation in the reaction-diffusion equations and its applications. Commun Nonlinear Sci Numer Simulat 2016;33:229-258.
  • [7] Pamuk S, G¨urb¨uz A. Stability analysis of the steady-state solution of a mathe-matical model in tumor angiogenesis. AIP Conference Proceedings 2004;729: 369 .
  • [8] Levine HA, Pamuk S, Sleeman BD, Hamilton MN. A mathematical model of capillary formation and development in tumour angiogenesis: penetration into the stroma. Bull. Math. Biol 2001;63:801-863.
  • [9] Pamuk S, Çay I. A 2D Mathematical Model for Tumor Angiogenesis: The Roles of Endothelials, Pericytes and Macrophages in the ECM. (in preperation).
  • [10] Karaoglu E, Merdan H. Hopf bifurcation of a ratio - dependent predator-prey model involving two discrete maturation time delays. Chaos, Solitons & Fractals 2014;68:159-168.
There are 10 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Serdal Pamuk

İrem Çay This is me

Publication Date March 31, 2018
Published in Issue Year 2018 Volume: 19 Issue: 1

Cite

APA Pamuk, S., & Çay, İ. (2018). STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, 19(1), 50-57. https://doi.org/10.18038/aubtda.323014
AMA Pamuk S, Çay İ. STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS. AUJST-A. March 2018;19(1):50-57. doi:10.18038/aubtda.323014
Chicago Pamuk, Serdal, and İrem Çay. “STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19, no. 1 (March 2018): 50-57. https://doi.org/10.18038/aubtda.323014.
EndNote Pamuk S, Çay İ (March 1, 2018) STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19 1 50–57.
IEEE S. Pamuk and İ. Çay, “STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS”, AUJST-A, vol. 19, no. 1, pp. 50–57, 2018, doi: 10.18038/aubtda.323014.
ISNAD Pamuk, Serdal - Çay, İrem. “STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 19/1 (March 2018), 50-57. https://doi.org/10.18038/aubtda.323014.
JAMA Pamuk S, Çay İ. STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS. AUJST-A. 2018;19:50–57.
MLA Pamuk, Serdal and İrem Çay. “STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, vol. 19, no. 1, 2018, pp. 50-57, doi:10.18038/aubtda.323014.
Vancouver Pamuk S, Çay İ. STABILITY AND HOPF BIFURCATION ANALYSIS OF A MATHEMATICAL MODEL IN TUMOR ANGIOGENESIS. AUJST-A. 2018;19(1):50-7.