Invasion speed

red ball How is the invasion speed of the fitter species modified by coupling to the profile?

"Competition on the edge of an expanding population,"

Daniel Swartz, Hyunseok Lee, M. Kardar & K. Korolev, arXiv:2301.07246 (2023) (off-line)

yellow ballMorphologies for isotropic growth with the standard FKPP competition:

yellow ball How rapidly does the mutant invade the wild type?

      

The faster of the uncoupled Fisher velocity, or circular arc velocity, determines invasion speed! 

yellow ball This simplicity is due to the density independent form of competition rule leading to a "pulled" Fisher waves.   

Different behavior occurs in a modified ("Allee") competitive model that leads to "pushed" Fisher waves.

       yellow ball Speed of the fitter species can be computed perturbatively (using the zeroth order profile, and a Cole-Hopf transformation),

and can actually be reversed by the growth advantage of the less fit species.

  

Quite generally, perturbation theory suggests that the front-expansion speed is not modified by coupling to the height profile for any pulled wave.

 

 

 

 

 

 

 

 

 

 

 

       yellow ball We may phenomenologically postulate that the invasion velocity is modified by the coupling to the profile as

Using the exact solution of the "Alee" modified Fisher equation as zeroth order,

a first order solution to the profile is obtained using the Cole-Hopf transformation,

the first order profile can then be used to compute a first order correction to the Fisher velocity in the above form.