Abstract
This chapter contains a general presentation of parabolic partial differential equations that are used in biology: Lotka-Volterra systems and chemical or enzymatic reactions. These are reaction-diffusion equations, or in a mathematical classification, semilinear equations. Our goal is to explain what mathematical properties follow from the set-up of the model: nonnegativity properties, monotonicity and entropy inequalities. We put a special emphasis on several general concepts: competitive or cooperative systems, law of mass action, derivation of the Michaelis-Menten law, Belousov-Zhabotinskii reaction. A connection is introduced between the heat equation and the Brownian motion.
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Perthame, B. (2015). Parabolic Equations in Biology. 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_1
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DOI: https://doi.org/10.1007/978-3-319-19500-1_1
Publisher Name: Springer, Cham
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