Fisher-Kolmogoroff Equation

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
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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[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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