Reaction-Diffusion Fronts in General Heterogeneous Media

4 Reaction-Diffusion Fronts in General Heterogeneous Media 

These methods have some limitations. Singular perturbation analysis is an effective tool if the solution is known to the leading order and if the reaction term is not given by KPP kinetics. The solution to the lowest order can be found for some particular non-KPP kinetic terms, but it is not known in general. This method requires, of course, that a small parameter is present in the model. It is necessary to assume that the spatial heterogeneities in the system introduce a small variation in the reaction 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

In this section we study the smooth heterogeneous problem 

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