cs.MAFeb 23, 2026

Agentic Hives: Equilibrium, Indeterminacy, and Endogenous Cycles in Self-Organizing Multi-Agent Systems

Authors: Jean-Philippe Garnier

Organizations: Br.AI.K · BrainiaK, Lille, France

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Abstract

Current multi-agent AI systems operate with a fixed number of agents whose roles are specified at design time. No formal theory governs when agents should be created, destroyed, or re-specialized at runtime-let alone how the population structure responds to changes in resources or objectives. We introduce the Agentic Hive, a framework in which a variable population of autonomous micro-agents-each equipped with a sandboxed execution environment and access to a language model-undergoes demographic dynamics: birth, duplication, specialization, and death. Agent families play the role of production sectors, compute and memory play the role of factors of production, and an orchestrator plays the dual role of Walrasian auctioneer and Global Workspace. Drawing on the multi-sector growth theory developed for dynamic general equilibrium (Benhabib & Nishimura, 1985; Venditti, 2005; Garnier, Nishimura & Venditti, 2013), we prove seven analytical results: (i) existence of a Hive Equilibrium via Brouwer's fixed-point theorem; (ii) Pareto optimality of the equilibrium allocation; (iii) multiplicity of equilibria under strategic complementarities between agent families; (iv)-(v) Stolper-Samuelson and Rybczynski analogs that predict how the Hive restructures in response to preference and resource shocks; (vi) Hopf bifurcation generating endogenous demographic cycles; and (vii) a sufficient condition for local asymptotic stability. The resulting regime diagram partitions the parameter space into regions of unique equilibrium, indeterminacy, endogenous cycles, and instability. Together with the comparative-statics matrices, it provides a formal governance toolkit that enables operators to predict and steer the demographic evolution of self-organizing multi-agent systems.

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Figure 1: Path dependence in the three-family Hive ( S=3S=3 , M=2M=2 ) with η1=1.3\eta_{1}=1.3 and γ12=γ21=0.4\gamma_{12}=\gamma_{21}=0.4 . (a) From N0=(5.0,5.0,5.0)\mathbf{N}_{0}=(5.0,5.0,5.0) , the system spirals into the perception-dominant morphology NA∗≈(6.1,5.8,1.4)\mathbf{N}^{*}_{A}\approx(6.1,5.8,1.4) . (b) From N0=(2.0,3.0,6.0)\mathbf{N}_{0}=(2.0,3.0,6.0) , it converges monotonically to the generation-dominant morphology NB∗≈(2.3,4.9,5.2)\mathbf{N}^{*}_{B}\approx(2.3,4.9,5.2) . The coexistence of two locally stable equilibria confirms Theorem 4.4 .
Figure 2: Endogenous demographic cycles (Hopf bifurcation) in the three-family Hive. Parameters: γ12=0.5\gamma_{12}=0.5 , γ21=−0.5\gamma_{21}=-0.5 (beyond the critical value γ21∗≈−0.42\gamma_{21}^{*}\approx-0.42 ). Starting from a perturbed equilibrium, the oscillation amplitude grows and saturates to a limit cycle with period T≈8.3T\approx 8.3 . The phase shifts between families produce the four “seasons” of the Hive: expansion, consolidation, contraction, and renewal.
Figure 3: Numerical regime diagram for the five-family Hive ( S=5S=5 , M=3M=3 ). Each marker corresponds to one parameter sweep point: ∙\bullet unique stable equilibrium, ■\blacksquare multiple stable equilibria, ▲\blacktriangle endogenous limit cycle, ×\times instability (boundary dynamics). Dashed lines indicate the critical boundaries γcrit≈0.20\gamma_{\mathrm{crit}}\approx 0.20 and ηcrit≈0.98\eta_{\mathrm{crit}}\approx 0.98 , confirming the four-region partition of Section 6 .
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