Transition between extinction and blow-up in a generalized Fisher–KPP model

• Stationary solutions of the Fisher–KPP equation are characterized and constructed.
• Criterions for blow-up, growth and extinction of solutions are developed.
• Stationary solutions are separatrices between blow-up and extinction phenomena.
• Examples are given that generalize previously known results.

Stationary solutions of the Fisher–KPP equation with general nonlinear diffusion and arbitrary reactional kinetic orders terms are characterized. Such stationary (separatrix-like) solutions disjoint the blow-up solutions from those showing extinction. In addition a criterion for general parameter values is presented, which allows determining the blow-up or vanishing character of the solutions.