Consider a nonlinear partial differential equation that represents the combined effects of diffusion and logistic population growth:
.
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:
.
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
![](img/desc29.png)
and boundary conditions
,
.
Here
is a dimensionless length for the diffusion zone.
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
is obtained for
and is shown in blue. The specified initial condition for the calculations is
.
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
. 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
.
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
).
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