Fisher-Kolmogoroff Equation

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Consider a nonlinear partial differential equation that represents the combined effects of diffusion and logistic population growth:

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.

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

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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Contributed by: Brian G. Higgins and Housam Binous  (November 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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.



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