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Spikes, Spots and Pulses

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Parabolic Equations in Biology

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Abstract

Typical behaviors of solutions of elliptic equations or systems are spikes or spots. In parabolic systems, solutions that vanish at both ends also occur and they travel with constant speed: we call them pulses. These types of solutions arise in various areas of biology: chemotaxis, adaptive evolution, FitzHugh-Nagumo system for electric pulses in nerves. We explain how this type of localized patterns can arise. A fascinating aspect is the emergence of unstable waves and dynamic patterns as in the Gray-Scott system and its many extensions.

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Authors and Affiliations

  1. Université Pierre et Marie Curie, Paris, France

    Benoît Perthame

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  1. Benoît Perthame
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Perthame, B. (2015). Spikes, Spots and Pulses. In: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-19500-1_5

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