Growth Morphology

red ballBeyond their relevance to the extinction transition, the couplings between shape of the front and population reproduction/mutation

have macroscopic manifestations in morphology of growing films:

yellow ball The parameter  λ  indicates the dependence of normal front velocity on slope:

For isotropic growth   λ= β1 =vo> 0,

       

Other growth rules (as in earlier model) lead to  λ< 0

   

The parameter  β1  indicates the in-plane front velocity on slope:

   

The parameters β2 and  c  determine the dependence of the extinction point on shape:

red ball   

yellow ball The growth morphology depends chiefly on (the signs of) λ , as well as the growth velocities g  and  α:   

  g  captures the reproductive fitness advantage; g=0 for netural (non-competitive) fitness

  α  indicates which species expands more rapidly (normal to the front), e.g. has larger size

Neutral evolution g=0 (equal fitness, no competition) leads to faceted patterns:

                                                         

           

       Competitive reproduction (fitter blue reproduces more rapidly) leads to the following patterns:

α < 0                                                                          α > 0

  λ< 0

  λ > 0

red ball Motivated by experiments on bacterial films, possible patterns for   λ> 0   were recently classified in:

yellow ball"Slow expanders invade by forming dented fronts in microbial colonies,"

Hyunseok Lee, J. Gore1 and K.S. Korolev. (2021 preprint)

Observed pattern is an example of competitive growth with λ> 0

 

 

Simulations of the stochastic PDEs lead to very similar patterns:

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