Contribution
Modern coexistence theory equates coexistence with positive invasion growth rates, without saying when signs suffice or which rates must be positive. This paper answers both for deterministic models: signs suffice when the invasion graph has no cycle, and then the rates that matter are species i’s at its −i communities, the subcommunities without i that no other missing species can invade.
Key insight
Coexistence is a statement about community assembly. Draw every subcommunity and every possible transition by invasion; if no sequence of invasions returns where it started, the boundary dynamics have a Morse decomposition, and permanence reduces to checking that the end-points of assembly without each species can be invaded by it.
Assumptions
- Equations (1): dx_i/dt = x_i f_i(z), dy/dt = g(z), with auxiliary variables y in a compact set (A1: local Lipschitz; A2: dissipative, a compact global attractor).
- A3a: for each ergodic measure µ on the extinction set, r_j(µ) ≠ 0 for every species j not supported by µ.
- A3b: r_j has the same sign at any two ergodic measures with the same support. It holds for Lotka–Volterra and replicator systems and whenever each face carries one invariant measure; it fails, for example, for two predators on one prey with a type II response (McGehee and Armstrong).
- Coexistence means robust permanence: permanence that survives small C⁰ perturbations of the vector field.
Key results
- Lemma 1. Resident species have r_i(µ) = 0.
- Definition. Invasion scheme: the signs r_i(S) for every proper subcommunity S. Invasion graph: S → T iff S ≠ T, r_j(S) > 0 for all j ∈ T \ S, and r_i(T) < 0 for all i ∈ S \ T.
- Theorem 1. A1–A3 and an acyclic invasion graph: robustly permanent if every subcommunity can be invaded by some species.
- Corollary 1. It suffices that r_i(S) > 0 at every −i community S. With C² vector fields, Remark 3 gives the converse: an uninvadable subcommunity has a stable set reaching the interior, so permanence fails.
- Corollary 2 and Proposition 1. Hofbauer’s average Lyapunov condition (positive weights p with Σ p_i r_i(µ) > 0 on all boundary measures) follows from signs alone iff the invasion scheme is sequentially permanent, a nested build-up of permanent communities, a strictly smaller class than acyclic graphs.
- Lotka–Volterra. With one interior equilibrium per face, r_i(S) = sgn((Ax*)_i + b_i), so the graph comes from linear algebra. Figure 1: two five-species competitive systems, both acyclic; one is robustly permanent, the other has nine −i communities, three uninvadable.
- Switching predator (§4.2): predator-mediated coexistence of bistable prey is possible, unlike a non-switching predator; the scheme is not sequentially permanent.
- Periodic chemostat (§4.3): numerically computed invasion graphs account for the loss of species 3 at a = 0.325, of species 2 at a = 0.275 and of species 1 and 2 at a = 0.2.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | With an acyclic invasion graph, signs of invasion growth rates decide robust permanence | strong | Theorem 1, Corollary 1, Remark 3; proofs read |
| C2 | Only species i’s rate at each −i community needs checking | strong | Corollary 1 |
| C3 | Average Lyapunov functions follow from signs only for sequentially permanent schemes | strong | Proposition 1, proof read |
| C4 | For stochastic models, acyclic graphs with invadable −i communities give stochastic persistence | conjecture | §5 |
Concepts
- Invasion growth rate r_i(µ): ∫ f_i dµ, the long-run per-capita growth of species i introduced at vanishing density into the dynamics of µ.
- −i community: a subcommunity without i in which every other missing species has a negative invasion growth rate. Found in practice by simulating from all species but i and removing the “atto-foxes”.
- Robust permanence: a compact interior attractor, persisting under small perturbations of the equations.
- Sequential permanence: an ordering of species such that each in turn has nonnegative rates at every remaining community.
Connections
It makes rigorous the “−i community” of Chesson (1994) and the recovery-from-rarity logic of Chesson’s framework (LIT-1219); its assembly graph is related to Serván and Allesina’s (2021) but allows multi-species invasions. Rock–paper–scissors (May and Leonard 1975) is the cyclic case where signs fail. None of these three is registered.
Bearing on the record
- QUESTION-132 defines a recovery exponent λ_i along the resident regime and takes λ_i > 0 as the mark of stabilizing coexistence. That λ_i is exactly r_i(µ) here. The reading adds two conditions the question should carry. (a) Positive recovery at one resident regime is not enough: it must be positive at every −i community, which may be a much smaller community than “all the others”. (b) Signs decide only if the invasion graph is acyclic. With cyclic replacement among operators, rock–paper–scissors among roles, all recovery exponents can be positive and coexistence still fail.
- QUESTION-083: the −i communities are the “early entrants shape which roles later ones can hold” structure the synthesis read into the paper (LIT, “Why this is here”). It is the right reading: −i communities are assembly end-points, and whether a role can be filled depends on the end-points reached without it.
- CLAIM-369, a question for the coordinator rather than a finding: in the symmetric, gradient case the objective L decreases along every trajectory, which should rule out heteroclinic cycles and so cyclic invasion graphs. If so, CLAIM-369’s class always satisfies this paper’s acyclicity hypothesis. The paper does not state this and the reading did not prove it.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- A labelling error in §4.2. The text says invasibility of {1, 2} and {1, 3} requires a/d > α and of {2, 3} requires a/d > 1 + α. From the stated equilibria, prey 2 invading {1, 3} (x1 = d/a, y = 1) has rate r(1 − αd/a), and prey 1 invading {2, 3} (x2 = d/a, y = 0) has the same rate, so both need a/d > α. The predator invading {1, 2} (x1 = x2 = 1/(1 + α), y = ½) has rate a/(1 + α) − d, so needs a/d > 1 + α. The two labels are swapped between {1, 2} and {2, 3}. The overall condition and the conclusion are unaffected.
- Other slips. “As the invasion graphs for α ≠ 0 are acyclic” probably means α ≠ 1; §4.3 gives β2 = −.3, the Figure 2 caption 0.3; in Appendix B and after Corollary 2, IG is written where IS is meant.
- Deterministic models only; A3b excludes common non-Lotka–Volterra cases; cyclic graphs are left open.
Open questions
- Does the stochastic conjecture of §5 hold?
- For cyclic invasion graphs, which quantitative information beyond signs decides permanence, and how should the rates then be decomposed into mechanisms?