Fisher-Kolmogoroff Equation
Fisher-Kolmogoroff Equation
Consider a nonlinear partial differential equation that represents the combined effects of diffusion and logistic population growth:
∂u
∂t
2
∂
∂
2
x
u
k
Reaction-diffusion equations have been widely applied in the physical and life sciences starting with the pioneering work of Roland Fisher who modeled the spread of an advantageous gene in a population [1]. His model equation, also known as the Fisher–Kolmogorov equation, has the following dimensionless form:
∂u
∂t
2
∂
∂
2
x
The Fisher–Kolmogorov equation is viewed as a prototype for studying reaction-diffusion systems that exhibit bifurcation behavior and traveling wave solutions.
The full mathematical statement of the transient problem is given by the preceding equation with initial condition
u(x,0)=g(x)
and boundary conditions
u(0,t)=0
u(L,t)=0
Here is a dimensionless length for the diffusion zone.
L
This Demonstration solves this transient reaction-diffusion problem and plots for equal to 0.005, 0.05, and 60, shown in red, green, and blue, respectively. The steady-state solution (x) is obtained for and is shown in blue. The specified initial condition for the calculations is .
u(x/L,t)
t
U
s
t=60
g(x)=0.4
A global stability analysis of the steady states shows that the critical value for occurs at . Thus for , the solution is the nontrivial one. For , diffusion stabilizes the steady-state solution =0. At the point we have a bifurcation to the nontrivial steady-state solution. This can be readily seen by selecting the versus plot. The red dot corresponds to the value of for the user-set value of .
L
L=
π
L>
π
L<π
U
s
L=π
U
s,max
L
U
s,max
L
Also shown is the phase plot for the steady solutions, namely, versus . The trajectories shown in the phase plot (blue, green, and magenta) are global solutions to the steady-state diffusion equation, but do not necessarily satisfy the boundary conditions. Nontrivial steady-state solutions that satisfy the imposed boundary conditions must lie within the light green region of the phase plot (defined by the orbit . For the user-specified parameters, the orbit shown in blue is a feasible nontrivial solution (i.e. for ).
U
s
U
s
{D,E,F}
{A,B,C}
L>π