Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model ${\partial }_{t}U=U\left\{f\left(U\right)-dV\right\},{\partial }_{t}V={\partial }_x\left\{f\left(U\right){\partial }_{x}V\right\}+rVf\left(V\right),$where $f\left(u\right)=1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal{U},\mathcal{V})$ the traveling wave profile and by $({\mathcal{U}}_{\pm },{\mathcal{V}}_{\pm })$ its asymptotic states at $\pm \infty$, we investigate existence in the regimes $d>1:\left({\mathcal{U}}_{-},{\mathcal{V}}_{-}\right)=\left(0,1\right)\text{and}\left({\mathcal{U}}_{+},{\mathcal{V}}_{+}\right)=\left(1,0\right),$$d<1:\left({\mathcal{U}}_{-},{\mathcal{V}}_{-}\right)=\left(1-d,1\right)\text{and}\left({\mathcal{U}}_{+},{\mathcal{V}}_{+}\right)=\left(1,0\right),$which are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter $c$ is strictly positive. We also derive an accurate approximation of the front profile in the singular limit $c\to 0$.

受癌症生物学中肿瘤生长的驱动，我们为简化的 Gatenby-Gawlinski 模型提供了对侵袭前沿的存在和不存在的完整分析 ${\partial }_{吨}你=你\left\{F\left(你\right)-d伏\right\},{\partial }_{吨}伏={\partial }_X\left\{F\left(你\right){\partial }_X伏\right\}+r伏F\left(伏\right),$在哪里 $F\left(你\right)=1-你$ 和参数 $d,r$是积极的。表示为$(\mathcal{你},\mathcal{伏})$ 行波剖面和由 $({\mathcal{你}}_{\pm },{\mathcal{伏}}_{\pm })$ 它的渐近状态在 $\pm \infty$, 我们调查制度中的存在 $d>1：\left({\mathcal{你}}_{-},{\mathcal{伏}}_{-}\right)=\left(0,1\right)\text{和}\left({\mathcal{你}}_{+},{\mathcal{伏}}_{+}\right)=\left(1,0\right),$$d<1：\left({\mathcal{你}}_{-},{\mathcal{伏}}_{-}\right)=\left(1-d,1\right)\text{和}\left({\mathcal{你}}_{+},{\mathcal{伏}}_{+}\right)=\left(1,0\right),$分别称为同质入侵和异质入侵。在这两种情况下，我们证明只要速度参数$C$是严格的正数。我们还推导出了奇异极限中前轮廓的准确近似值$C\to 0$.