Phenomenological Derivation of the Reaction-Diffusion Equation

Let 5 be an arbitrary surface enclosing a time-independent volume V. The general law of conservation for the particle density states that the rate of change of the amount of particles in V is due to the flow of particles across the surface S plus the net production of particles in the volume V  

Since the integration volume is arbitrary, we obtain the conservation equation, or continuity equation, for p  

The continuity equation (2.7) needs to be closed via a constitutive equation for the flux J. If fhe fransporf process corresponds fo classical diffusion, the constitutive equation is given by Pick s first law. 

If D is constant, (2.9) reduces to (2.3) in the one-dimensional case. In some models for insect populations, models for bacterial chemotaxis, or for animal grouping due to social aggregation, the diffusion coefficient can be an explicit function of the particle density or a function of other chemical components. 

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