Hyperbolic limit for a biological invasion

In a spatially heterogeneous environment the propagation speed of biological invasion varies in space. The traveling wave theory homogeneous case is not extended to case. Taking singular limit hyperbolic scale good way study such with constant speed. goal this project understand effect diffusion on spatial environment. For purpose, we consider \begin{document}$ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $\end{document} where $ m nonconstant carrying capacity, s {U\over m} starvation measure and \gamma(s) s^{\tilde{k}}, \tilde{k} \ge 1 $. driven diffusion. We show that even if nonconstant.