Nullcline analysis of the general well-mixed SMC model.

Stable equilibria are shown as filled circles and unstable equilibria as open circles. Arrows indicate the direction of population change on the nullclines. (A) The P nullcline (solid red line) lies above the B nullcline (blue dashed line), leading to exclusion of beneficials. (B) Because the P nullcline is steeper than the B nullcline, if the nullclines cross their intersection (open circle) is a saddle and therefore unstable. The stable manifold of the saddle divides the interior of the quadrant into the sets of initial points leading to competitive dominance by one type of microbe and competitive exclusion of the other. (C) The B nullcline lies above the P nullcline, leading to exclusion of pathogens. (D) Small rates of microbial immigration are permitted, with the qualitative effect of preventing complete extinction when interactions within the mucus layer would otherwise lead to competitive exclusion.