Diffusion modeling of tumor-CD4$ ^+ $-cytokine interactions with treatments: asymptotic behavior and stationary patterns

Wenbin Yang; Yujing Gao; Xiaojuan Wang

doi:10.3934/dcdsb.2021090

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2022, Volume 27, Issue 3: 1285-1300. Doi: 10.3934/dcdsb.2021090

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Diffusion modeling of tumor-CD4$ ^+ $-cytokine interactions with treatments: asymptotic behavior and stationary patterns

1.
School of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi 710121, China
2.
Department of Gastroenterology, Xi'an Honghui Hospital, Xi'an, Shaanxi 710000, China

^* Corresponding author: Wenbin Yang
^* Corresponding author: Wenbin Yang

Received: October 2020

Revised: February 2021

Early access: March 2021

Published: March 2022

The first author is supported by NSF of China grant 12001425

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Abstract

In this work, we consider a diffusive tumor-CD4$ ^+ $-cytokine interactions model with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria, including the system persistence and the stability conditions. We also give the existence of nonconstant positive steady states (i.e., a stationary pattern), which indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems, linearization method, the method of energy integral and the Leray-Schauder degree.

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Mathematics Subject Classification: Primary: 35B40, 35B36; Secondary: 35K57.

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