Table 3. Summary of results.

Competition structure	I	II	III
Producer-only stability	$\theta >\frac{{\beta }_0}{{\beta }_1}$	$\theta >\frac{{\beta }_0}{{\beta }_1}$	$\theta >\frac{{\beta }_0}{{\beta }_1}$
Invasion of 2° producers	$\omega <\frac{{\beta }_0}{{\beta }_1}$	β0 > β1	$\theta <\frac{{\beta }_0}{{\beta }_1}$
Invasion of global producers	μ < 1	false	false
Cheater-only stability	false	false	false
Invasion of 2° producers	true (ϕ < 1)	true (ϕ < 1)	r01 > r10
Invasion of global producers	true (ψ < 1)	true (ψ < 1)	true (ϕ < 1)
Coexistence stability	[messy]	[messy]	[messy]
Invasion of 2° producers	β1(ϕθ − ω) + β0(ωθ − ϕ − θ2 + 1) > 0	$\frac{\varphi +\theta (\theta -1)-1}{\varphi \theta -1}>\frac{{\beta }_1}{{\beta }_0}$	false
Invasion of global producers	β0(μθ − ψ) + β1(ψθ − μ − θ2 + 1) > 0	(ψ − θ)(θβ1 − β0) > 0	(ϕ − θ)(β1 θ − β0) > 0

A comparison of the invasion conditions at and stability conditions of each quasi-equilibrium for each competition structure. The conditions for stability of coexistence are omitted because they are mathematically intractable, though numerical analysis showed stability can be easily achieved for various parameter combinations. It is assumed that at the producer-only and coexistence quasi-equilibria, R >> k while at the cheater-only quasi-equilibrium, k >> R.