Fisher-Kolmogoroff Equation

​
dimensionless length for diffusion zone
3.3
transient and steady-state solutions
Consider a nonlinear partial differential equation that represents the combined effects of diffusion and logistic population growth:
∂u
∂t
=
2
∂
u
∂
2
x
+ru1-
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
∂
u
∂
2
x
+u(1-u)
.
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
L
is a dimensionless length for the diffusion zone.
This Demonstration solves this transient reaction-diffusion problem and plots
u(x/L,t)
for
t
equal to 0.005, 0.05, and 60, shown in red, green, and blue, respectively. The steady-state solution
U
s
(x)
is obtained for
t=60
and is shown in blue. The specified initial condition for the calculations is
g(x)=0.4
.
A global stability analysis of the steady states shows that the critical value for
L
occurs at
L=
π
. Thus for
L>
π
, the solution is the nontrivial one. For
L<π
, diffusion stabilizes the steady-state solution
U
s
=0
. At the point
L=π
we have a bifurcation to the nontrivial steady-state solution. This can be readily seen by selecting the
U
s,max
versus
L
plot. The red dot corresponds to the value of
U
s,max
for the user-set value of
L
.
Also shown is the phase plot for the steady solutions, namely,

U
s
versus
U
s
. 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
{D,E,F}
. For the user-specified parameters, the orbit
{A,B,C}
shown in blue is a feasible nontrivial solution (i.e. for
L>π
).

References

[1] R. A. Fisher, "The Wave of Advance of Advantageous Genes," Ann. Eugenics, 7, 1937 pp. 353–369.
[2] P. Grindrod, The Theory and Application of Reaction-Diffusion Equations: Patterns and Waves, Oxford: Clarendon Press, 1996.
[3] J. D. Murray, Mathematical Biology I. An Introduction, 3rd ed., New York: Springer, 2002.
[4] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed., New York: Springer, 2003.

Permanent Citation

Brian G. Higgins, Housam Binous
​
​"Fisher-Kolmogoroff Equation"​
​http://demonstrations.wolfram.com/FisherKolmogoroffEquation/​
​Wolfram Demonstrations Project​
​Published: November 3, 2011