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We analyze the transition between pulled and pushed fronts both analyticallyand numerically from a model-independent perspective. Based on minimalconceptual assumptions, we show that pushed fronts bifurcate from a branch ofpulled fronts with an effective speed correction that scales quadratically inthe bifurcation parameter. Strikingly, we find that in this general contextwithout assumptions on comparison principles, the pulled front loses stabilityand gives way to a pushed front when monotonicity in the leading edge is lost.Our methods rely on far-field core decompositions that identify explicitlyasymptotics in the leading edge of the front. We show how the theoreticalconstruction can be directly implemented to yield effective algorithms thatdetermine spreading speeds and bifurcation points with exponentially smallerror in the domain size. Example applications considered here include anextended Fisher-KPP equation, a Fisher-Burgers equation, negative taxis incombination with logistic population growth, an autocatalytic reaction, and aLotka-Volterra model.

Pushed-to-pulled front transitions: continuation, speed scalings, and hidden monotonicity