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.