Parabolic Equations in Biology

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.