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

<p style='text-indent:20px;'>In this work, we consider a diffusive tumor-CD4<inline-formula><tex-math id="M2">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula>-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.</p>