Contribution
It turns community assembly into a graph problem and finds a class in which the graph is simple. Vertices are the sub-communities that can coexist; an edge S → S′ is an invasion by one species, at low density, that moves the community from S’s equilibrium to S′‘s. Two ways of assembling are compared: species arriving one at a time with the community settling in between (“bottom-up”), and all species introduced at once (“top-down”). The chapter proves that the two reach the same end-points when the graph is acyclic and the attractors are exactly the sub-communities no absent species can invade, and shows that symmetric competitive Lotka–Volterra communities meet both conditions.
Key insight
A directed graph is acyclic exactly when some quantity increases along every edge (Lemma 1). In symmetric competition that quantity is MacArthur’s Lyapunov function, which at an equilibrium is the total biomass when r = v. So assembly is an ascent: each successful invasion raises V, no sequence of invasions can return to an earlier community, and the end-points are the local maxima of V on the feasible set. Order of arrival can still matter, but only by choosing among several maxima, and only when the interaction matrix is not positive definite.
Assumptions
- Invasions are rare (§3.2.2): the community reaches its asymptotic state between invasions. This rules out the rock–paper–scissors community’s three-species state, which needs a third invader before the second has finished excluding the first.
- Invaders arrive at low density, so only local stability of a resident state matters.
- Fixed-point attractors, with Condition 1 of the appendix: each feasible state is globally attractive within its species set, and an invasion succeeds iff the invader grows when rare at the resident’s (time-averaged) state.
- The model class (§3.2.3, §3.5.3). ẋ_i = v_i x_i(γ_i − Σ_j b_ij x_j), from A = D(v)BD(w) by x_j = w_j N_j and γ_i = r_i/v_i; r_i > 0, v_i, w_i > 0; B symmetric, b_ii = 1, all entries positive (“Assume that all parameters are positive”); every principal submatrix of B nonsingular, and (B^S | γ^S) of full column rank. The last two are genericity conditions; they exclude singular interaction matrices.
- Top-down assembly is the attractor reached from all species present at an arbitrary initial condition.
Key results
- Lyapunov function (3.4), (3.10)–(3.11). V(x) = 2Σγ_i x_i − Σ x_i x_j b_ij, with dV/dt = 2Σ v_i x_i(γ_i − Σ_j b_ij x_j)² ≥ 0 for any symmetric B, positive definite or not (MacArthur 1970). At an equilibrium V(x*) = Σγ_i x_i*, the total biomass when r = v.
- Attractors (§3.5.3). Every attractor is a constrained local maximum of V: an equilibrium x^S that no absent species can invade, γ_k − Σ_j b_kj x_j^S < 0 (3.12), with B^(S) positive definite, from the Hessian −2B^(S) (3.14). Conversely such an x^S is a local maximum (3.15).
- Theorem 1 (§3.5.2). Under Conditions 1 and 2 (attractors are exactly the uninvadable sub-communities), (a) if the graph restricted to each feasible S is acyclic, the graph is accessible, and each S can be assembled from its own species with no extinctions; (b) if the whole graph is acyclic, bottom-up and top-down assembly have the same end-points.
- B positive definite (§3.3). The graph is accessible, has no directed cycles, and has a unique sink, the unique global attractor; “no historical contingency is possible”, and when r = v the end-point is the feasible sub-community of greatest total biomass.
- B not positive definite (§3.3). “As in the stable case, the existence of V implies that G is acyclic.” The graph is still accessible and acyclic, its sinks are the attractors of top-down assembly, and the two processes are equivalent, but there are several end-points and the one reached depends on the invasion sequence (Figure 3.3: three end-points for five species).
- Consumer–resource models (§3.5.4). With every resource kept present, the consumers’ assembly strictly raises Marsland et al.’s ψ(Y) = ½Σ(Y_k − r_k)², which orders the states, so the graph is acyclic. If consumers can drive resources extinct, ψ falls when a resource reinvades, and assembly graphs with cycles exist (Figure 3.5).
- Faster invasion (§3.3.1). With invasions as a Poisson process of rate ρ, the fraction of sub-communities assembly can reach falls as ρ grows, toward the top-down attractors alone (simulated, Figure 3.4).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Bottom-up and top-down assembly share end-points when the assembly graph is acyclic and attractors are the uninvadable sub-communities | strong | Theorem 1, Lemmas 1–2, proofs read |
| C2 | For symmetric competitive Lotka–Volterra the assembly graph is acyclic, whether or not B is positive definite | strong, one step implicit | MacArthur’s V (3.10)–(3.11); the step from V to edges is asserted and checks (see below) |
| C3 | With B positive definite there is one end-point, reachable from the empty community, and every feasible sub-community can be built without extinctions | strong | §3.3 with Theorem 1(a) and Proposition 1 |
| C4 | With B indefinite there are several end-points, so priority effects remain | moderate | §3.3, one five-species example (Figure 3.3) |
| C5 | Consumer–resource assembly is acyclic while every resource persists, and can cycle when resources can be excluded | moderate | §3.5.4 argument; cycles shown by example (Figure 3.5) |
| C6 | Acyclicity is sufficient, not necessary, for equivalence | weak | stated in §3.4 with a pointer to an appendix example that is not clearly marked |
Method
Enumerate the feasible sub-communities (at most 2ⁿ), compute each one’s equilibrium and each absent species’ growth rate when rare, draw an edge for each successful invasion to the community the augmented system settles in, and read accessibility, sinks and cycles off the graph. The chapter notes that construction is exponential in n and is meant for small communities.
Concepts
- Assembly graph G: vertices are the feasible sub-communities (a globally stable interior equilibrium for those species, and ∅); an edge S → S′ labelled j when j ∉ S invades x^S when rare and the community settles in S′.
- Accessible: every vertex has a path from ∅ (an “assembly path”).
- Assembly end-point: a strongly connected set of vertices whose invasions stay inside it and which is a sink of the condensed graph; a single sink vertex in the simplest case.
- Assembly cycle: a directed cycle of communities, the generalization of intransitive competition.
- Equivalence: the bottom-up end-points are exactly the top-down attractors.
- Perfect addition ordering: an assembly path for S along which no species goes extinct.
Connections
The assembly-graph construction comes from Law and Morton (1993), Hang-Kwang and Pimm (1993) and Capitán, Cuesta and Bascompte (2009); the cycling example is Schreiber and Rittenhouse (2004). The Lyapunov function is MacArthur’s (1970), and the consumer–resource ordering is Marsland, Cui and Mehta’s minimum environmental perturbation principle (LIT-1230, NOTE-1352). The chapter’s graph differs from Hofbauer and Schreiber’s invasion graph (LIT-1233, NOTE-1379), which it does not cite. There, an edge S → T needs every species in T but not S to invade S and every species lost to be unable to invade T. Here an edge is one invasion followed by the dynamics. Bunin (LIT-1234, NOTE-1397) gives the multiple-attractor phase of random symmetric Lotka–Volterra models in which the chapter’s indefinite case lives.
Bearing on the record
- CLAIM-369, the note of 2026-10-29’s open point. The triage asked whether this paper shows that the symmetric case gives an acyclic invasion graph and order-independent assembly. It shows the first for its own assembly graph, and the second only when the interaction matrix is positive definite. For CLAIM-369’s class, A = UᵀU and r = Uᵀb − c, the claim’s objective is the chapter’s V up to sign and a constant. With v = w = 1, L(n) = ½‖b‖² − (rᵀn − ½nᵀAn), and at any equilibrium nᵀAn = rᵀn, so L(n*) = ½‖b‖² − ½V(n*). Each successful invasion therefore lowers L, and since A = UᵀU is positive semidefinite, it is either positive definite (U of full column rank) or singular. When it is positive definite, the chapter gives one end-point, reached in any order of arrival, and every feasible sub-community assembled without extinctions. When it is singular, which CLAIM-369 allows (“With A singular there may be several”), the chapter’s genericity assumption fails and it says nothing. The chapter’s indefinite case, with acyclicity but priority effects, cannot arise from a Gram matrix. Two further limits on the transfer: the chapter needs positive entries in B, which is CLAIM-369’s sign condition for the ecological reading, and its pool admits only species with r_i > 0.
- Hofbauer and Schreiber’s graph, closed here by derivation. The chapter’s “the existence of V implies that G is acyclic” is asserted. It holds, and it holds for the invasion graph of LIT-1233 too. At equilibria x_S and x_T of any symmetric system, V(x_S) = rᵀx_S and V(x_T) = rᵀx_T, and writing f_k(S) = r_k − (Ax_S)k for a species’ growth when rare at x_S, V(x_T) − V(x_S) = Σ{j ∈ T∖S} f_j(S) x_{T,j} − Σ_{i ∈ S∖T} f_i(T) x_{S,i}, because the cross terms cancel when A is symmetric. Hofbauer and Schreiber’s edge S → T requires f_j(S) > 0 for j ∈ T∖S and f_i(T) < 0 for i ∈ S∖T, so both sums make the difference positive: V strictly increases along every edge of their invasion graph, and the graph is acyclic. This needs only symmetry and an equilibrium for each vertex, not positive definiteness or positive entries. The same identity with a single invader gives the chapter’s step. Checked by a record script (2026-10-29) on 300 random systems of three to six species, Gram, signed-indefinite and positive-entry symmetric matrices: 3,274 feasible equilibria, 14,266 invasion-graph edges, V increased along every one and the identity held to 10⁻⁹. So CLAIM-369’s class always meets LIT-1233’s acyclicity hypothesis, and signs of invasion growth rates decide robust permanence there. Filed as THEORY-1168 (Active); a dated note on CLAIM-369 is proposed.
- QUESTION-083, entrenchment by order of arrival. In the potential case with a positive definite Gram matrix, there is none: every order of arrival ends in the same community, and no invasion sequence returns to an earlier one. Order-dependence needs a non-positive- definite symmetric matrix, which a Gram matrix is not, or asymmetry. This agrees with Marsland, Cui and Mehta’s monotone assembly (the question’s note of 2026-10-29) and sharpens it: in this class an early entrant can be displaced, but the community it is displaced into is fixed by the pool, not by history.
- QUESTION-132, “judged by invasion graphs”. In the potential case the invasion graph is always acyclic (above), so the question’s second added condition, that signs decide only if the graph is acyclic, is met automatically there; cycles among roles need asymmetry or a resource that can be excluded (§3.5.4).
- The triage’s expectation. The web summary the File tier rested on (“ends in the same communities whatever the order”) is right only for positive definite B. The paper does show the graph acyclic, so by the owner’s call it stays filed.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The step from V to acyclicity is not written out. Lemma 1 needs an h that increases along edges; §3.3 asserts that V is one. The step holds (above), but the chapter does not prove it.
- Genericity excludes singular matrices, which are common in consumer–resource reductions with fewer resources than consumers.
- The slow-invasion regime is essential. Equivalence is asymptotic, and the number of invasions needed to reach an end-point can be very large (§3.4).
- The indefinite case is shown by one example, and the probability of reaching each end-point is left open.
- Slips. Main-text equations (3.1) and (3.2) are typeset with the sum “ij” where j is meant; “top-assembly and assembly” (§3.3) drops “bottom-up”; “enpoint” in Figure 3.1’s caption; (3.13) sums over j ≠ k while (3.12) sums over all j, which agree only because x_k^S = 0. The “example in Appendix” that acyclicity is not necessary is not clearly identified; Figure 3.5’s left panel, with a cycle and an accessible top-down attractor, may be it.
- Version. The dissertation chapter, not the journal article, was read.
Open questions
- With A = UᵀU singular (more components than features), how are the equilibria of a vertex chosen, and does the invasion graph stay acyclic once vertices are sets of equilibria?
- When B is indefinite, what is the probability that random invasion sequences end at each end-point (§3.4’s random walk on G)?