On a PDE-ODE-PDE model for two interacting cell populations under the influence of an acidic environment and with nonlocal intra- and interspecific growth limitation

TL;DR

The study introduces a two-phenotype tumor model with active density $u$, quiescent density $w$, and proton concentration $h$, incorporating nonlinear diffusion, nonlocal intra- and interspecific competition via convolutions with kernels $J_1(x,h)$ and $J_2(x,h)$, and acidity-triggered phenotype switching. A rigorous analysis develops an approximate problem with regularization, proves local and then global existence for the approximate system using a fixed-point framework and a priori estimates (including a Moser iteration), and then passes to the limit to obtain a bounded, nonnegative weak solution of the original problem. Complementary 1D simulations explore boundedness and pattern formation as functions of the exponents $(\alpha,\beta,\gamma)$ and kernel choices, revealing blow-up thresholds $\alpha^*$ that depend sensitively on kernel type (e.g., $\alpha^*\approx6.2$ for logistic and $\alpha^*\approx14.7$ for uniform) and acidity-driven interactions. The work provides a mathematically rigorous foundation for nonlocal PDE-ODE-PDE models of interacting tumor cell populations in acidic microenvironments and highlights how nonlocality and environmental feedback shape aggregation and invasion dynamics, with implications for understanding tumor heterogeneity and therapy strategies.

Abstract

We consider a model for the dynamics of active cells interacting with their quiescent counterparts under the influence of acidity characterized by proton concentration. The active cells perform nonlinear diffusion and infer proliferation or decay, according to the strength of spatially nonlocal intra- and interspecific interactions. The two cell phenotypes are interchangeable, the transitions depending on the environmental acidity. We prove global existence of a weak solution to the considered PDE-ODE-PDE system and perform numerical simulations in 1D to informally investigate boundedness and patterning behavior in dependence of the system's parameters and kernels involved in the nonlocal terms.

Figures (6)

• Figure 1: Initial conditions $u_0$, $w_0$, $h_0$.
• Figure 2: Simulation results of model \ref{['IBVP']} with $J_1=J_2=J_L$, i.e. logistic kernels, $\beta=\gamma=\mu_1=1$, $\alpha =2,5,6.1,6.2$ (columns from left to right, respectively). Component $u$ of solution starts to become unbounded near $\alpha =6.2$. In the rightmost column a blow-up occurs in the next time step.
• Figure 3: Simulation results of model \ref{['IBVP']} with $J_1=J_2=J_U$, i.e. uniform kernels, $\beta=\gamma=\mu_1=1$, $\alpha =2,6.2,14.6,14.7$ (columns from left to right, respectively). Component $u$ of solution starts to become unbounded near $\alpha =14.7$. In the rightmost column a blow-up occurs in the next time step.
• Figure 4: Simulations of models \ref{['IBVP']} and \ref{['modelww']} with $\alpha =2$, $\mu_1=1$, and (from left to right) $\beta =1,10, 100,1000$ and $\gamma =1000$ in column 1, $\gamma =1$ in columns 2-4. First row: $J_1=J_2=J_U$; 2nd row: model without $w$, with $J=J_U$; 3rd row: $J_1=J_2=J_L$; 4th row: model without $w$, with $J=J_L$
• Figure 5: Simulations of model \ref{['modelww']} without $w$ with $\beta = \mu = 1$. Columns 1 and 2: $J$ logistic and $\alpha = 4.4,4.5$. Columns 3 and 4: $J$ uniform and $\alpha = 5.7,5.8$. In the 2nd and 4th column a blow-up occurs in the next time step.

Theorems & Definitions (33)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Remark 3.4
• Theorem 3.5
• proof
• Lemma 3.6