  We analyze the transition between pulled and pushed fronts both analytically
and numerically from a model-independent perspective. Based on minimal
conceptual assumptions, we show that pushed fronts bifurcate from a branch of
pulled fronts with an effective speed correction that scales quadratically in
the bifurcation parameter. Strikingly, we find that in this general context
without assumptions on comparison principles, the pulled front loses stability
and gives way to a pushed front when monotonicity in the leading edge is lost.
Our methods rely on far-field core decompositions that identify explicitly
asymptotics in the leading edge of the front. We show how the theoretical
construction can be directly implemented to yield effective algorithms that
determine spreading speeds and bifurcation points with exponentially small
error in the domain size. Example applications considered here include an
extended Fisher-KPP equation, a Fisher-Burgers equation, negative taxis in
combination with logistic population growth, an autocatalytic reaction, and a
Lotka-Volterra model.
