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Volume 3, Issue 3
Turing and Hopf Bifurcation in a Diffusive Tumor-Immune Model

Jingnan Wang & Shengnan Liu

DOI: 10.12150/jnma.2021.477

J. Nonl. Mod. Anal., 3 (2021), pp. 477-493.

Published online: 2022-06

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.

  • Keywords

Tumor-immune model, Diffusion, Hopf bifurcation, Turing bifurcation, Stability.

  • AMS Subject Headings

34C23, 35K57

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-3-477, author = {Wang , Jingnan and Liu , Shengnan}, title = {Turing and Hopf Bifurcation in a Diffusive Tumor-Immune Model}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2022}, volume = {3}, number = {3}, pages = {477--493}, abstract = {

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2021.477}, url = {http://global-sci.org/intro/article_detail/jnma/20679.html} }
TY - JOUR T1 - Turing and Hopf Bifurcation in a Diffusive Tumor-Immune Model AU - Wang , Jingnan AU - Liu , Shengnan JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 477 EP - 493 PY - 2022 DA - 2022/06 SN - 3 DO - http://doi.org/10.12150/jnma.2021.477 UR - https://global-sci.org/intro/article_detail/jnma/20679.html KW - Tumor-immune model, Diffusion, Hopf bifurcation, Turing bifurcation, Stability. AB -

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.

Wang , Jingnan and Liu , Shengnan. (2022). Turing and Hopf Bifurcation in a Diffusive Tumor-Immune Model. Journal of Nonlinear Modeling and Analysis. 3 (3). 477-493. doi:10.12150/jnma.2021.477
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