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

Yıl 2018, Cilt: 19 Sayı: 1, 50 - 57, 31.03.2018

Serdal Pamuk İrem Çay

https://doi.org/10.18038/aubtda.323014

Öz

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.

Anahtar Kelimeler

Equilibrium solution, Hopf bifurcation

Kaynakça

• [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.