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Hyperbolic limit for a biological invasion

  • *Corresponding author: Hyunjoon Park

    *Corresponding author: Hyunjoon Park

Dedicated to the memory of Professor Masayasu Mimura

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  • In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider

    $ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $

    where $ m $ is a nonconstant carrying capacity, $ s = {U\over m} $ is a starvation measure and $ \gamma(s) = s^{\tilde{k}}, \tilde{k} \ge 1 $. The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if $ m $ is nonconstant.

    Mathematics Subject Classification: Primary: 35B25, 35G31, 35K57, 82C24; Secondary: 92D25.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  A linearly connected patch system is viewed as a general diffusion in $ {{\bf R}}^1 $

    Figure 2.  Numerical simulation of (7) and (IP). The black line indicates the curve which moves according to (IP) and other colors indicates the values of the function $ u^{\varepsilon} = U^{\varepsilon}/m $. The yellow color indicates that the solution is close to 1 and the blue color indicates that the solution is equal to 0

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