nucleation

CLAIM-546: Negative frequency dependence that keeps two types in coexistence does not make their interaction cooperative: in Hawk–Dove with cost C above value V each type increases when rare and they coexist at the interior equilibrium of V/C Hawks, with Hawks still taking the whole resource from Doves, and on the Black Queen account producers of a leaked good and the cells that use it without paying coexist at a frequency set by the community's need; so whether types persist together and whether they cooperate are separate questions, and a coexistence criterion answers only the first

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Properties8
StatusProposed
Rolethesis
Warrantdemonstrated
Tagsgame-theory, ecology, altruism
Linedistributed-agency, formal-apparatus
Bears onQUESTION-132 — Under what conditions do initially equivalent adaptive operators differentiate into functionally complementary specialists that each have positive recovery growth when rare, rather than into roles kept in place by external load balancing or by persistent authority asymmetries?
GroundsTHEORY-1048 — On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required, THEORY-567 — In the donation game, where a cooperator pays c for another to get b, defectors win in a well-mixed population, and each of five mechanisms makes cooperators evolutionarily stable above a benefit-to-cost threshold: kin selection r > c/b, direct reciprocity w > c/b, indirect reciprocity q > c/b, network reciprocity b/c > k and group selection b/c > 1 + n/m; the rules cover altruism only, not mutualism, coordination or conflict, CASE-118 — Two competing components with interaction matrix A = [​[1, ρ], [ρ, 1]]: with equal growth rates their invasion growth is 1 − ρ, the smallest eigenvalue of the Gram matrix, and with growth rates 1 ± δ the same matrix supports coexistence only when |δ| < (1 − ρ)/(1 + ρ), so feature geometry alone does not fix coexistence, LIT-1219 — Mechanisms of Maintenance of Species Diversity
ComplementsCLAIM-547 — Cooperation within groups and the coexistence of groups are distinct targets that can trade off: in the group-selection rule of Nowak's five, b/c > 1 + n/m, cooperation spreads because a group that grows to size n splits and another group is driven extinct, so the mechanism that stabilizes cooperation inside groups works by removing groups; cooperation local to a group, the persistence of several groups whose interests are opposed, and concern for those outside one's group are three achievements, and none of them secures the next

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The claim

Coexistence theory’s test is mutual invasibility: each type must increase when rare (Chesson, LIT-1219; CASE-118 for two competitors). Evolutionary game theory’s test for cooperation is whether cooperators resist defectors (THEORY-567). The two tests can come apart.

Hawk–Dove. Contests over a resource of value V. A Hawk escalates, a Dove displays and retreats, and an injury costs C > V. Payoffs to the row player:

HawkDove
Hawk(V − C)/2V
Dove0V/2

With a fraction p of Hawks, a Hawk earns p(V − C)/2 + (1 − p)V and a Dove earns (1 − p)V/2. Their difference is (V − pC)/2.

  • Both types increase when rare. As p → 0 the difference tends to V/2 > 0, so Hawks invade. As p → 1 it tends to (V − C)/2 < 0, so Doves invade.
  • They coexist at p = V/C*, where the difference is zero. That is coexistence by negative frequency dependence, in exactly coexistence theory’s sense.
  • Exploitation is intact. Every Hawk–Dove meeting at p* gives the Hawk V and the Dove 0. The mixture is stable, and the interaction it stabilizes is not cooperative.

The Black Queen account (THEORY-1048) gives the same thing in microbes. Cells that lose a costly function whose product leaks are favoured until the remaining producers just supply the community. The loss is frequency dependent and stops short, so producers and beneficiaries coexist. Most cells come to depend on the few, “with no relatedness, cooperation or collective required”.

So a criterion for whether types persist together does not settle whether they cooperate. It settles whether either can exclude the other.

What it does not say

  • That coexistence and cooperation exclude each other. Mutualisms coexist and cooperate. The claim is only that the first does not imply the second.
  • That Nowak’s rules are coexistence criteria. They state when cooperators beat defectors (THEORY-567), which can mean cooperators replace defectors, not that both persist.
  • Anything about mutualism and facilitation stabilizing coexistence. The synthesis’s §2 says positive interactions can stabilize coexistence. The record holds no work on that, and the work the synthesis cites there could not be identified (curation entry of 2026-11-04).

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