A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion

A number of important phenomena in ecology can be modeled by one-dimensional, nonlinear reaction-diffusion PDEs. This paper considers a modified Fisher PDE for which the diffusion term is nonlinear. A nonstandard finite difference scheme is constructed using methods generated by the previous work of Mickens. As a check on the mathematical properties of this scheme, a linear stability analysis is carried out for the two fixed-points appearing in the differential and difference equations. The finite difference scheme is shown to have solutions which satisfy a positivity condition as well as the requirement of boundedness. Further, the scheme is explicit and a functional relationship is obtained between the space and time step-sizes. A numerical test of the scheme is done for a particular initial/boundary value problem. A brief discussion of how the work can be extended and/or generalized is also given.