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

A full isocline analysis of the competition models yields zero-growth surfaces in (J, C, U)-space. An approximate analysis in (C,U)-space can be conducted by modeling juvenile canopy-former cover implicitly, characterizing J as a function of C and U. To do this, we provisionally fix adult canopy cover and understory cover, set Eqs. 3b, 4b, and 5b equal to zero, and solve for J:

	(A.1a)
	(A.1b)
,	(A.1c)

where the subscripts S, R, and H denote standoff, reversal, and hierarchical competition, respectively. We can then substitute  for J in the equations for adult canopy-former and understory species dynamics, yielding two-equation models for adult canopy former and understory species cover that approximate the full three-equation models (cf. Alexander and Roughgarden 1996; Connolly and Roughgarden 1999).  While not a comprehensive graphical representation of the full model, the qualitative patterns indicated—the conditions necessary for the existence of a coexistence equilibrium and its stability—are consistent with full three-dimensional analyses.

In the hierarchical model, the understory population cannot exclude the canopy former, and there are only two possible outcomes of competition: exclusion of the understory species (Fig. A.1A), or coexistence (Fig. A.1B).  For standoff and reversal competition, the potential outcomes of competition depend critically upon the relationship between canopy former growth ability ( ) and canopy former mortality rate ( ).  When , the understory population cannot exclude the canopy former.  This result is not immediately obvious because, in contrast to the hierarchical model, canopy former dynamics depend on U (Eqs. 3, 4), so the canopy former isocline is not horizontal.  However, it can be shown analytically.  For standoff competition, the canopy former's zero-growth isocline can be rearranged as follows:

.

(A.2)

 If the understory species can exclude the canopy former, then the canopy former isocline, the right-hand side of Eq. A.2, must decline to zero at some point. If C = 0, then = 0 (Eq. A.1a). Thus, Eq. A.2 becomes

.

(A.3)

If 0, this cannot be true, so the understory species cannot exclude the canopy former. (A similar proof applies for the reversal case.)  Thus, as with hierarchical competition, the only possible outcomes of competition are exclusion of the understory species (Fig. A.1C), or coexistence (Fig. A.1D).  By contrast, when canopy growth ability does not exceed mortality rate (), exclusion of the canopy-former by the understory species becomes possible.  As a result, potential outcomes of competition include either species excluding the other, stable coexistence, or priority effect (Fig. A.2).

FIG. A.1.  Zero-growth isoclines illustrating possible outcomes of hierarchical competition (A, B), and standoff and reversal competition when (C, D).  Dotted lines indicate the adult canopy former isoclines; solid lines indicate the understory species isoclines.  The possible outcomes of competition for these cases are exclusion of the understory species by the canopy former (A, C), or coexistence (B, D).  Arrows indicate the direction of population trajectories in each of the regions bounded by the isoclines.

FIG. A.2.  Zero-growth isoclines illustrating possible outcomes of standoff and reversal competition when . (A) Competitive exclusion of the understory species by the canopy former. (B) Competitive exclusion of the canopy former by the understory species. (C) Stable coexistence. (D) Priority effect (unstable coexistence equilibrium). Arrows indicate the direction of population trajectories in each of the regions bounded by the isoclines.

Literature cited

Alexander, S. A., and J. Roughgarden. 1996. Larval transport and population dynamics of intertidal barnacles: a coupled benthic/oceanic model. Ecological Monographs 66:259–275.

Connolly, S. R., and J. Roughgarden. 1999. Theory of marine communities: competition, predation, and recruitment-dependent interaction strength. Ecological Monographs 69:277–296.