Competition & Invasion

red ball Competitive advantage of one a species leads to invasion into neighboring territory (as a uniform front if stochasticity is ignored):

yellow ball Invasion speed: The proportion,  f=0, of the invading species can be modeled by a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation:

red ball Traveling wave solutions are broadly classified as [reviewed in W. van Saarloos, Phys. Rep 386, 29 (2003)]

yellow ballPulled waves are controlled by the leading edge, with speed obtained by linearization around  f=0  as

yellow ballPushed waves are influenced by the the entire profile, their velocity is not obtained to linearization.

An exactly solvable case (triggered wave) is provided by

which is used to model the "Alee effect" with a minimum population size required for growth.

The exact form of the wave profile in this model is given by