Tumor-immune cell interactions by a fully parabolic chemotaxis model with logistic source
Rafael Díaz Fuentes
TL;DR
This work analyzes a fully parabolic chemotaxis system modeling tumor-immune interactions with logistic growth for immune cells and a chemical mediator, described by $u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu_1u^k-\mu_2u^{k+1}$, $v_t=\Delta v+\alpha w-\beta v-\gamma uv$, and $w_t=\Delta w-\delta uw+\mu_3 w(1-w)$ in a bounded domain with Neumann boundaries. The authors prove global existence and boundedness of classical solutions in dimensions $n\ge 3$, distinguishing the two regimes $k>1$ and $k=1$: for $k>1$ boundedness holds under parameter-based conditions independent of initial data, while for $k=1$ a computable lower bound on $\mu_2$ that depends on $n$, $\chi$, initial data, and other parameters is required. The proofs hinge on carefully constructed a priori estimates built around an energy functional $\Phi(t)=\int u^p+\int|\nabla v|^{2p}+\int|\nabla w|^{2p}$, gradient bounds, and a extensibility criterion linking $L^p$ bounds of $u$ to global existence. The results extend prior two-equation models and show robustness to large chemotactic sensitivity, with explicit computable constants provided. This contributes to the mathematical understanding of tumor-immune dynamics and supports long-time qualitative analyses, including potential convergence towards carrying capacity and depletion of chemical signals.
Abstract
This work studies the existence of classical solutions to a class of chemotaxis systems reading \[\begin{cases} u_t = Δu-χ\nabla\cdot(u \nabla v) + μ_1 u^k -μ_2 u^{k+1}, & \text{in} \; Ω\times(0,T_{\text{max}}), \\ v_t= Δv+αw-βv-γu v, & \text{in} \; Ω\times(0,T_{\text{max}}), \\ w_t= Δw-δu w+ μ_3 w(1-w), & \text{in} \; Ω\times(0,T_{\text{max}}), \\ \frac{\partial u}{\partialν}=\frac{\partial v}{\partialν}=\frac{\partial w}{\partialν}=0, & \text{on} \; \partialΩ\times(0,T_{\text{max}}), \\ u(x,0)=u_0(x), \quad v(x,0)= v_0(x), \quad w(x,0)= w_0(x), & x\in\overlineΩ, \end{cases}\] that model interactions between tumor (i.e., $w$) and immune cells (i.e., $u$) with a logistic-type source term $μ_1 u^k - μ_2 u^{k+1}$, $k\geq1$, also in presence of a chemical signal (i.e., $v$). The model parameters $χ, μ_1,μ_2, μ_3, α, β, γ$, and $δ$ are all positive. The value $T_{\text{max}}$ indicates the maximum instant of time up to which solutions are defined. Our focus is on examining the global existence in a bounded domain $Ω\subset \mathbb{R}^n, n \geq 3$, under Neumann boundary conditions. We distinguish between two scenarios: $k>1$ and $k=1$. The first case allows to prove boundedness under smaller assumptions relying only on the model parameters instead of on the initial data, while the second case requires an extra condition relating the parameters $χ, μ_2$, $n$, and the initial data $\lVert v_0 \rVert_{L^\infty(Ω)}$. This model can be seen as an extension of those previously examined in [11] and [4], being the former a system with only two equations and the latter the same model without logistic.