Traveling Waves in the Spatial 1D Fisher-KPP Equation With Allee Effect

Abstract A natural question is how changing the diffusion or reaction terms in the equation changes the structure of the traveling wave solution of the Fisher-KPP equation. This paper answers the question of how the addition of a nonlinear term expressing the Allee effect changes the classification of information about the existence and shape of the traveling wave. In addition to the details of the information for the front-type traveling wave solution, the existence and characterization of traveling waves that are unbounded at the endpoints of a finite interval are precisely analyzed. The results for this classification are obtained by the two-dimensional phase space including to infinity, which corresponds to the dynamics of a two-dimensional ordinary differential equation satisfied by the traveling wave. These are derived by the Poincaré compactification, one of the compactifications of phase space. Based on the results of the Fisher-KPP reaction-diffusion equation for traveling waves in the linear diffusion case, the author’s previous work can be integrated to discuss the effect of the Allee effect on traveling waves. In addition, a comparison and discussion of how the solution structure of the traveling wave changes with changes in the diffusion and reaction terms is given.