Phases of the model

 Coexistence of  "all" species requires sufficiently large spatio-temporal noise.

 To gain some understanding of this phase behavior, replace the random sum    with the number 

 With this assumption, the coupled equations reduced to independent forms:

We already encountered the corresponding steady distribution of the above, and can check against numerical results:

Check for the stability of a given set of average populations  :

The above approximations suggest:  :

This is supported by simulations:

 How this modifies May's instability argument?

 In the limit of large   , we find    , and linear stability analysis yields:

 Eigenvalues remain negative and stable coexistence is possible for

Next (Summary) Outline Growth & saturation Mean field analysis Extinction on a seascape Growth on a seascape Diversity vs Stability Coexistence on a seascape Phase behavior Summary