Sean R. Connolly and Soyoka Muko. 2003. Space pre emption, size-dependent competition, and the coexistence of clonal growth forms. Ecology 84:2979–2988.

Define the critical understory longevity of species i, , as the understory longevity at which species i's invasibility criterion is just met. Then the isoclines in Fig. 1 plot the critical understory longevities of the understory species ( ,) and the canopy former ( ) as functions of understory colonization ability. The transition from coexistence to priority effect occurs where the invasibility thresholds cross; i.e., where . A priority effect occurs when two conditions are met:  (1)  (i.e., the canopy former's invasibility threshold lies below the understory species' in Fig. 1); and (2)  (the understory species can persist in the absence of the canopy former).  If we combine these conditions, and rearrange, we obtain the following expression for the existence of a priority effect:

(B.1)

for standoff competition and

(B.2)

for reversal competition.  K_C and K_J represent the single-species equilibrium cover of adult and juvenile canopy formers, respectively. The point where these inequalities are just satisfied are the points where invasibility isoclines cross in Fig. 1D,E.  It is clear from inequality B.1 that, under standoff competition, the transition from coexistence to priority effect is increasingly favored as understory life history shifts towards low spatial turnover (as and decline, a broader range of canopy former parameters can satisfy the inequality). This is apparent in the shift from coexistence to priority effect as one moves towards lower understory space colonization ability and greater longevity (leftwards and upwards in Fig. 1D,E). Since appears in the right-hand side of inequality B.2, a similar proof cannot be offered for the reversal case.  However, extensive numerical investigations indicate that the transition from coexistence to priority effect occurs with decreasing understory spatial turnover in this model as well.