THEORY-1168: In a Lotka–Volterra community with a symmetric interaction matrix, the growth-weighted biomass rᵀx strictly increases along every edge of the invasion graph, so the invasion graph has no cycles

Source

Serván and Allesina (2021), LIT-1824, read in NOTE-1422, for the assembly-graph result and MacArthur’s Lyapunov function; Hofbauer and Schreiber (2022), LIT-1233, read in NOTE-1379, for the invasion graph.

What was actually shown

Take ẋ_i = x_i(r_i − (Ax)_i) with A symmetric. (Per-capita weights v_i > 0, as in Serván and Allesina’s ẋ_i = v_i x_i(γ_i − (Bx)_i), change no sign and leave the argument as it is, with γ for r.) For a set of species S with an equilibrium x_S supported on S, write f_k(S) = r_k − (Ax_S)_k, the growth of species k when rare at x_S; f_k(S) = 0 for k ∈ S.

1. At an equilibrium, x_Sᵀ(r − Ax_S) = 0, so MacArthur’s V(x) = 2rᵀx − xᵀAx takes the value V(x_S) = rᵀx_S.
2. For two equilibria, because A is symmetric the cross terms cancel: f(S)ᵀx_T − f(T)ᵀx_S = rᵀx_T − x_SᵀAx_T − rᵀx_S + x_TᵀAx_S = rᵀ(x_T − x_S).
3. Dropping the zero terms, 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}.
4. Hofbauer and Schreiber’s edge S → T requires S ≠ T, f_j(S) > 0 for every j ∈ T∖S, and f_i(T) < 0 for every i ∈ S∖T. Both sums then make the difference strictly positive, so rᵀx rises along every edge and no directed cycle exists.
5. Serván and Allesina’s edge, a single invader j that grows at x_S and a community that settles at x_T, is covered the same way: V rises when j is added at small density, since ∂V/∂x_j = 2f_j(S) > 0, and V does not fall along the flow, so V(x_T) > V(x_S).

The check that could have failed: a script enumerated every feasible equilibrium of 300 random symmetric systems and every invasion-graph edge between them, and compared rᵀx across each edge and against the identity of step 3. No edge violated it.

When A = UᵀU and r = Uᵀb − c (CLAIM-369), V(x*) = ‖b‖² − 2L(x*), so the claim’s objective L falls along every invasion-graph edge.

What this does not say

• Not that assembly is order-independent. Acyclicity removes cycles, not alternative end-points. A symmetric matrix that is not positive definite has several uninvadable communities, and which one assembly reaches depends on order of arrival (Serván and Allesina’s Figure 3.3). Uniqueness needs A positive definite, as a Gram matrix of independent vectors is.
• Not that the invasion graph is defined. The argument needs one equilibrium per vertex. With singular principal submatrices, as for a Gram matrix with more species than features, a vertex can carry a continuum of equilibria, and the statement is about whichever are chosen.
• Not competition. Neither positive entries nor positive definiteness is used, so it holds for symmetric facilitation too; the ecological reading as competition is a further condition.
• Not beyond symmetry. With A asymmetric the cross terms do not cancel, and cyclic invasion graphs exist (rock–paper–scissors).