Reaction–diffusion system

Reaction–diffusion systems are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local <a href="/facts/Chemical_reaction/GpqWlKrz">chemical reactions</a> in which the substances are transformed into each other, and <a href="/facts/Diffusion/KbAxroBa">diffusion</a> which causes the substances to spread out over a surface in space.
Reaction–diffusion systems are naturally applied in <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in <a href="/facts/Biology/1M1ee3pB">biology</a>, <a href="/facts/Geology/WGvVRIYF">geology</a> and <a href="/facts/Physics/a7aH2y08">physics</a> (neutron diffusion theory) and <a href="/facts/Ecology/YE8upuJU">ecology</a>. Mathematically, reaction–diffusion systems take the form of semi-linear <a href="/facts/Parabolic_partial_differential_equation/gJQfRUlr">parabolic partial differential equations</a>. They can be represented in the general form

∂
          
            t
          
        
        
          q
        
        =
        
          
            
              D
              _
            
            _
          
        
        
        
          ∇
          
            2
          
        
        
          q
        
        +
        
          R
        
        (
        
          q
        
        )
        ,
      
    
    {\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),}

where q(x, t) represents the unknown vector function, D is a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> of <a href="/facts/Diffusion_coefficient/GSngNem0">diffusion coefficients</a>, and R accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of <a href="/facts/Travelling_wave/I3EeyInV">travelling waves</a> and wave-like phenomena as well as other <a href="/facts/Self-organization/NSGBKxUa">self-organized</a> <a href="/facts/Pattern_formation/HLpWMVt1">patterns</a> like stripes, hexagons or more intricate structure like <a href="/facts/Dissipative_soliton/xpAu7sEE">dissipative solitons</a>. Such patterns have been dubbed "<a href="/facts/Turing_pattern/aI05dj4m">Turing patterns</a>". Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.

Reaction–diffusion system open-in-new

Reaction–diffusion system