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On a non-local equation arising in population dynamics

Abstract : We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits
Keywords : traveling fronts waves
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Jerome Coville, Louis Dupaigne. On a non-local equation arising in population dynamics. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Cambridge University Press (CUP), 2007, 137 (4), pp.727-755. ⟨10.1017/S0308210504000721⟩. ⟨hal-02659746⟩

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