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The long time behavior of the solution to a nonlinear diffusion problem in population genetics and combustion

, : The long time behavior of the solution to a nonlinear diffusion problem in population genetics and combustion. Journal of Theoretical Biology 104(4): 473-484

The Fisher or Kolmogoroff-Petrovsky-Piscounoff equation exemplifies wave-like phenomena occurring in population genetics and combustion. An extension of this equation was proposed and closed form traveling wave, stationary and symmetric solutions were obtained. Employing the transformation properties of the extended equation, 2 integral invariants for the problem are obtained and 2 Lyapunov functionals, which characterize the evolution of the profile to a uniformly propagating traveling wave, are constructed. A generalization of this modified Fisher equation is proposed and its integral invariants, traveling wave solutions and wave speeds were obtained, as well as the Lyapunov functionals which describe its asymptotic evolution.

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