Fisher-KPP equation belongs to the class of
reaction–diffusion equations: in fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term : f(u,x,t) = r u (1-u),\, which can exhibit traveling wave solutions that switch between equilibrium states given by f(u) = 0. Such equations occur, e.g., in
ecology,
physiology,
combustion,
crystallization,
plasma physics, and in general
phase transition problems. Fisher proposed this equation in his 1937 paper
The wave of advance of advantageous genes in the context of
population dynamics to describe the spatial spread of an advantageous
allele and explored its travelling wave solutions. For every wave speed c \geq 2 \sqrt{r D} ( c \geq 2 in dimensionless form) it admits travelling
wave solutions of the form : u(x,t)=v(x \pm ct)\equiv v(z),\, where \textstyle v is increasing and : \lim_{z\rightarrow-\infty}v\left( z\right) =0,\quad\lim_{z\rightarrow\infty }v\left( z\right) =1. That is, the solution switches from the equilibrium state
u = 0 to the equilibrium state
u = 1. No such solution exists for
c c=\pm 5/\sqrt{6}, all solutions can be found in a closed form, with : v(z) = \left( 1 + C \mathrm{exp}\left(\mp{z}/{\sqrt6}\right) \right)^{-2} where C is arbitrary, and the above limit conditions are satisfied for C>0. Proof of the existence of travelling wave solutions and analysis of their properties is often done by the
phase space method. ==KPP equation==