Self-similar solutions to a density-dependent reaction-diffusion model.

In this paper, we investigated a density-dependent reaction-diffusion equation, u(t)=(u(m))(xx)+u-u(m). This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation, which is widely used in population dynamics, combustion theory, and plasma physics. By employing a suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, the connection between the self-similar solution and the traveling wave solution is quite clear from these results. Moreover, the solutions were found in a manner that propagates either to the right or to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. Applications to spatiotemporal pattern formation in biological populations is discussed.