Effective carrying capacity and analytical solution of a particular case   of the Richards-like two-species population dynamics model
  We consider a generalized two-species population dynamic model and analytically solve it for the amensalism and commensalism ecological interactions. These two-species models can be simplified to a one-species model with a time dependent extrinsic growth factor. With a one-species model with an effective carrying capacity one is able to retrieve the steady state solutions of the previous one-species model. The equivalence obtained between the effective carrying capacity and the extrinsic growth factor is complete only for a particular case, the Gompertz model. Here we unveil important aspects of sigmoid growth curves, which are relevant to growth processes and population dynamics. 
Brenno Caetano Troca Cabella, Fabiano Ribeiro and Alexandre Souto   Martinez
q-bio.PE cond-mat.dis-nn physics.bio-ph