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Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

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Abstract:

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, ) and is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b := b(a) ∈ (, ∞), and furthermore, b(ยท) : (0, ) → (, ∞) is continuous and bijective.

Document Type: Research Article

Publication date: 2005-08-01

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