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

Jean-Philippe Garnier

Year: 2026Venue: arXiv preprintArea: cs.MAType: PreprintEmbeddings: 55

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 87%

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Summary

The paper introduces the 'Agentic Hive' framework, a theoretical model for self-organizing multi-agent systems with variable populations. It maps agent demographics to multi-sector economic growth theory, proving results such as the existence of a Hive Equilibrium, Pareto optimality, and conditions for endogenous cycles via Hopf bifurcation. The framework uses an orchestrator as a Walrasian auctioneer to manage resource allocation among agent families, providing a governance toolkit for predicting and steering system evolution.

Entities (13)

Jean-Philippe Garnier · person · 99%Agentic Hive · framework · 98%Orchestrator · component · 95%Hive Equilibrium · concept · 92%Agent Families · entity · 90%BrainiaK · organization · 90%Hopf Bifurcation · mathematical-concept · 88%Rybczynski · theorem · 85%Stolper-Samuelson · theorem · 85%AutoGen · software-system · 80%CrewAI · software-system · 80%MetaGPT · software-system · 80%Swarm · software-system · 80%

Relation Signals (11)

Agentic Hive → proposedby → Jean-Philippe Garnier

confidence 95% · The paper introduces the Agentic Hive... Jean-Philippe Garnier

Orchestrator → functionsas → Walrasian auctioneer

confidence 92% · an orchestrator plays the dual role of Walrasian auctioneer and Global Workspace

Agentic Hive → exhibits → Hive Equilibrium

confidence 90% · existence of a Hive Equilibrium via Brouwer's fixed-point theorem

Agentic Hive → uses → Orchestrator

confidence 90% · an orchestrator plays the dual role of Walrasian auctioneer and Global Workspace

Agentic Hive → exhibits → Hopf Bifurcation

confidence 88% · Hopf bifurcation generating endogenous demographic cycles

Agentic Hive → applies → Stolper-Samuelson

confidence 85% · Stolper-Samuelson and Rybczynski analogs that predict how the Hive restructures

Agentic Hive → applies → Rybczynski

confidence 85% · Stolper-Samuelson and Rybczynski analogs

AutoGen → relatedto → Multi-agent LLM systems

confidence 80% · Systems such as AutoGen [ 29]... enable multi-agent collaboration with LLMs

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Abstract

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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arXiv:2603.00130v2 [cs.MA] 27 Apr 2026 Agentic Hives: Equilibrium, Indeterminacy, and Endogenous Cycles in Self-Organizing Multi-Agent Systems Jean-Philippe Garnier ∗ BrainiaK Lille, France February 2026 Abstract Current multi-agent AI systems operate with a fixed number ofagents 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 theAgentic 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—undergoesdemographic dynamics: birth, duplica- tion, 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 equi- librium (Benhabib & Nishimura, 1985; Venditti, 2005; Garnier, Nishimura & Ven- ditti, 2013), we prove seven analytical results: (i) existence of a Hive Equilibrium via Brouwer’s fixed-point theorem; (i) Pareto optimality of the equilibrium alloca- tion; (i) 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 bifurca- tion generating endogenous demographic cycles; and (vii) asufficient condition for local asymptotic stability. The resultingregime diagrampartitions 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. Keywords:multi-agent systems, general equilibrium, dynamic general equilib- rium, agent demographics, indeterminacy, endogenous cycles, Hopf bifurcation, LLM orchestration. ∗ Corresponding author. E-mail:jeanphi.garnier@brainiak.tech. 1 Contents 1 Introduction3 1.1 The missing theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Our approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Relation to prior work by the author. . . . . . . . . . . . . . . . . . . . . 4 2 Related work5 3 The Agentic Hive model5 3.1 Agents, families, and resources. . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Production and externalities. . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Social welfare and the orchestrator’s problem. . . . . . . . . . . . . . . . . 7 3.4 Population dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.5 Hive Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Main results9 4.1 Existence of Hive Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Pareto optimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Multiplicity and indeterminacy. . . . . . . . . . . . . . . . . . . . . . . . 10 4.4 Stolper–Samuelson analog. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.5 Rybczynski analog. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.6 Endogenous cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.7 Stability conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Numerical illustrations13 5.1 Three-family Hive (S= 3,M= 2). . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Five-family Hive (S= 5,M= 3). . . . . . . . . . . . . . . . . . . . . . . 15 6 The regime diagram17 7 Discussion18 7.1 A macroeconomic governance framework. . . . . . . . . . . . . . . . . . . 18 7.2 Connection to Global Workspace Theory. . . . . . . . . . . . . . . . . . . 18 7.3 Imperfect information and mechanism design. . . . . . . . . . . . . . . . . 18 7.4 Relation to evolutionary dynamics. . . . . . . . . . . . . . . . . . . . . . . 19 7.5 Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8 Conclusion20 A Proofs and technical details22 A.1 Proof details for Theorem 4.4 (Multiplicity). . . . . . . . . . . . . . . . . 22 A.2 Proof details for Theorem 4.6 (Stolper–Samuelson). . . . . . . . . . . . . 23 A.3 Proof details for Theorem 4.10 (Endogenous cycles). . . . . . . . . . . . . 23 A.4 Lyapunov property and welfare monotonicity. . . . . . . . . . . . . . . . . 24 2 1 Introduction The past two years have witnessed an explosion of multi-agent systems built around large language models (LLMs). Systems such as AutoGen [ 29], CrewAI [5], MetaGPT [14], and Swarm [24] enable several LLM-powered agents to collaborate on complex tasks. All of these systems share a common architectural assumption:the number of agents is fixed at design time, and their roles are pre-assigned. The system designer decides that there will be a “planner” agent, a “coder” agent, and a “reviewer” agent, then hard-codes the interaction protocol. If the task mix changes, or if resources become scarce, or if one agent family becomes redundant, the system has no principled mechanism for adaptation. This is the equivalent of running an economy without a theoryof growth: you can allo- cate resources across existing firms, but you have no formal basis for deciding when new firms should enter, when existing ones should exit, or how theindustrial structure should respond to a technology shock. 1.1 The missing theory We argue that the missing ingredient is amacroeconomic theory of agent demographics. Specifically, we need formal answers to four questions: (Q1)Entry:When should new agents be created, and of what type? (Q2)Exit:When should existing agents be removed? (Q3)Specialization:How should undifferentiated agents acquire functional roles? (Q4)Restructuring:How does the entire population structure respond to changesin resources or objectives? No existing framework provides analytical answers to (Q1)–(Q4). Swarm intelligence [ 7, 18] studies self-organization but provides no equilibrium theory. Multi-agent reinforcement learning (MARL) [19] learns policies but does not model population dynamics. Agent- based computational economics (ACE) [ 26] simulates agent populations but offers no closed-form results. 1.2 Our approach We observe that (Q1)–(Q4) arepreciselythe questions addressed by the theory of multi- sector economic growth—a branch of dynamic general equilibrium theory that has been developed over seven decades, from Arrow & Debreu [ 1], Debreu [4], and McKenzie [20] to Garnier, Nishimura & Venditti [ 10]. The mapping is direct: 3 Multi-sector economyAgentic Hive Production sectorsAgent families (perception, planning,. . . ) Factors (capital, labor)Resources (GPU, memory, attention, . . . ) Output of sectorjTask completion by familyj Factor pricesShadow prices of resources Consumer preferencesSystem objectives (quality, cost, latency, . . . ) Walrasian auctioneerOrchestrator / Global Workspace Inter-sectoral externalities Cross-family spillovers Steady-state populationHive Equilibrium This is not a metaphor. As we show formally in Section3, the mathematical objects are identical: the population dynamics of agent families are governed by a dynami- cal system whose steady states are general equilibria, whose comparative statics obey Stolper–Samuelson and Rybczynski theorems, and whose stability properties determine the existence of endogenous demographic cycles. 1.3 Contributions We introduce theAgentic Hive, a formal framework for self-organizing multi-agent systems with variable population, and establish the following results: 1.Existence(Theorem 4.1): Under standard regularity assumptions, at least one Hive Equilibrium exists. 2.Pareto optimality(Theorem 4.2): Every Hive Equilibrium is Pareto-optimal when the orchestrator has full information. 3.Multiplicity(Theorem 4.4): Under strategic complementarities between agent fam- ilies, multiple equilibria coexist—corresponding to distinct “morphologies” of the Hive. 4.Stolper–Samuelson analog(Theorem 4.6): A magnification effect links prefer- ence changes to resource price changes. 5.Rybczynski analog(Theorem 4.8): A magnification effect links resource endow- ment changes to population restructuring. 6.Endogenous cycles(Theorem 4.10): A Hopf bifurcation generates periodic demo- graphic oscillations—the “seasons” of the Hive. 7.Stability(Theorem 4.12): A sufficient condition on mortality rates guarantees local asymptotic stability. Together, these results yield aregime diagram(Section 6) that partitions the parameter space into regions of qualitatively distinct behavior, providing a formal governance toolkit for designers and operators of multi-agent systems. 1.4 Relation to prior work by the author The author’s doctoral work [ 9,10] characterized local indeterminacy in continuous-time multi-sector growth models, establishing conditions under which competitive equilibrium cycles arise from the interaction of returns to scale and consumer preferences. The present paper transports this toolkit to the entirely different domain of multi-agent AI systems, 4 extending it from fixed-population general equilibrium to variable populations with en- dogenous demographics and multi-regime dynamics. 2 Related work Multi-agent LLM systems.AutoGen [29], MetaGPT [14], CrewAI [5], and Swarm [24] enable multi-agent collaboration with LLMs. All assume a fixed agent set with pre-defined roles. Recent surveys [ 12] catalog the rapid growth of this field but do not identify popu- lation dynamics as a research gap. Swarm intelligence.Ant Colony Optimization [ 7], Particle Swarm Optimization [18], and related algorithms use local interaction rules to achieve collective behavior. These ap- proaches do not provide equilibrium concepts, comparativestatics, or stability theorems. Evolutionary game theory.The replicator equation [ 13] governs population shares under frequency-dependent fitness. Our population dynamics (Section 3.4) are a general- ization of replicator dynamics to absolute populations with endogenous resource allocation and multi-sector externalities. Agent-based computational economics.ACE [ 26] simulates heterogeneous agent populations but produces no closed-form results. Our framework provides analytical the- orems that complement simulation. Multi-sector growth theory.Benhabib & Nishimura [ 3] established the existence of competitive equilibrium cycles in multi-sector optimal growth models. Venditti [ 28] and Garnier, Nishimura & Venditti [10] extended these results to characterize indeterminacy under various assumptions on returns to scale and preferences. We transport this entire toolkit to agent demographics. Global Workspace Theory.Baars [ 2] proposed that consciousness arises from a global workspace that integrates and broadcasts information fromspecialist processors. Our or- chestrator plays an analogous role: it selects, integrates, and broadcasts resource alloca- tion signals across agent families. 3 The Agentic Hive model 3.1 Agents, families, and resources Definition 3.1(Agentic Hive).AnAgentic Hiveis a tupleH= (A t ,F,R,w,Γ,Θ) where: •A t is the set of agents alive at timet, with|A t |=N total (t) variable; •F=1, . . . , Sis a finite set ofagent families(functional specializations); •R= (R 1 , . . . , R M )∈R M + is the vector of resource endowments; •w= (w 1 , . . . , w S )∈∆ S is the vector of global preferences over family outputs; • Γ = (γ jk ) j,k ∈R S×S is the externality matrix; 5 • Θ is the orchestrator. We denote byN j t ∈R + the population of familyjat timet, and byN t = (N 1 t , . . . , N S t )∈ R S + the population vector. Definition 3.2(Agent).An agenta∈A t is a tuplea= (g a , σ a , f a , μ a , τ a , ω a , ζ a ) where: •g a ∈G: genome (base prompt, policy, configuration); •σ a ∈Σ: internal state (local memory, context); •f a ∈F ∪∅: family assignment (∅for stem agents); •μ a ∈R K + : performance profile (Kmetrics); •τ a ∈R + : age; •ω a : sandboxed execution environment; •ζ a ∈[0,1]: degree of specialization (0 = totipotent, 1 = fully specialized). Definition 3.3(Stem agent).An agentais astem agentifζ a = 0 andf a =∅. A stem agent canduplicate(create a copy with reset state) orspecialize(transition to a family j∈Fwithζ a >0). Resources are organized intoMtypes. Each agent in familyjconsumes a resource vector; the aggregate consumption by familyjof resourcemis denotedK m j . The matrix K= (K m j ) j,m ∈R S×M + records the full resource allocation. Assumption 1(Budget).Total population is bounded by a budget constraint: P S j=1 c j N j ≤ B, wherec j >0 is the per-agent maintenance cost of familyjandB >0 is the total budget. Define thefeasible population set: Ω = n N∈R S + : P S j=1 c j N j ≤B o .(1) Ω is compact and convex. 3.2 Production and externalities Each family produces output that depends on its resource allocation, its population, and spillovers from other families. Definition 3.4(Sectoral production).The output of familyjis: Y j (K j , N j ,N −j ) =A j ·F j (K j )·(N j ) η j ·Γ j (N −j ),(2) where: •A j >0 is the total factor productivity of familyj; •F j :R M + →R + is a CES production function in resources: F j (K j ) = " M X m=1 α jm K m j ρ j −1 ρ j # ρ j ρ j −1 ,(3) withα jm >0, P m α jm = 1, andρ j >0 the elasticity of substitution; 6 • (N j ) η j captures returns to scale within the family, withη j >0; • Γ j (N −j ) = Q k6=j (N k ) γ jk is the externality term, withγ jk ∈R. Assumption 2(Production regularity).For eachj: (i)F j isC 2 , increasing, and strictly concave; (i) Inada conditions hold:∂F j /∂K m j → ∞asK m j →0 + and∂F j /∂K m j →0 asK m j → ∞; (i) 0< η j <1 (decreasing returns within family; relaxed toη j ≥1 in Section 4.3). Remark3.5.The assumptionη j <1 guarantees that the marginal value of an additional agent diminishes within each family, ensuring interior equilibria. In Theorem 4.4, we relax this toη j ≥1 for some families, which is the source of indeterminacy. The externality matrixE= (γ jk ) is the central structural object of the theory: •γ jk >0: familykgenerates a positive spillover on familyj(complementarity); •γ jk <0: familykimposes a negative externality on familyj(congestion); •γ jk = 0: independence. 3.3 Social welfare and the orchestrator’s problem Definition 3.6(Social welfare).The social welfare function is: W(K,N) = S X j=1 w j ·u Y j (K j , N j ,N −j ) − S X j=1 c j N j ,(4) whereu:R + →Ris a CRRA utility:u(Y) =Y 1−σ /(1−σ) forσ >0,σ6= 1, and u(Y) = lnYforσ= 1. Definition 3.7(Orchestrator’s problem).Given populationN, the orchestrator solves theinner problem: W ∗ (N) = max K≥0 W(K,N) s.t. S X j=1 K m j ≤R m ∀m∈1, . . . , M.(5) Assumption 3(Utility regularity).uisC 2 , strictly increasing, and strictly concave onR ++ . Assumption 4(Weak externalities).The externality matrix satisfies kEk ∞ = max j X k6=j |γ jk |< 1−η max 1 +σ η max ,(6) whereη max = max j η j <1 (from Assumption 2(i)) andσis the CRRA parameter. Remark3.8.Assumption4ensures that the value functionW ∗ (N) is globally concave on Ω. The bound arises because the Hessian ofW ∗ with respect toNhas diagonal terms of orderη j −1<0 (diminishing returns) and off-diagonal terms of orderγ jk (externalities). Concavity holds when the negative diagonal dominates the off-diagonal perturbation. This assumption is relaxed in Section 4.3to obtain indeterminacy. Under Assumptions 2and3, the inner problem (5) has a unique solutionK ∗ (N) = (K m∗ j (N)) j,m , with associated shadow pricesλ ∗ (N) = (λ 1∗ , . . . , λ M∗ ) satisfying the first- 7 order conditions: w j u ′ (Y ∗ j )·A j ∂F j ∂K m j K ∗ ·(N j ) η j Γ j =λ m∗ ∀j, m.(7) Definition 3.9(Marginal social value).The marginal social value of an agent in familyj is: V j (N) = dW ∗ dN j =w j u ′ (Y ∗ j )· ∂Y j ∂N j K ∗ −c j ,(8) where the equality follows from the envelope theorem (the terms involving∂K ∗ /∂N j vanish at the optimum). Explicitly: V j (N) =w j u ′ (Y ∗ j )·A j F j (K ∗ j )η j (N j ) η j −1 Γ j (N −j )−c j .(9) 3.4 Population dynamics The demographic dynamics of the Hive are governed by a systemof ODEs: Definition 3.10(Population dynamics).The population of familyjevolves according to: dN j dt = Φ j (N) =V j (N)·N j ,j= 1, . . . , S.(10) This is aselection dynamic[ 13]: families with positive marginal value (V j >0) grow; families with negative marginal value (V j <0) decline. In vector form: ̇ N= Φ(N), where Φ :R S + →R S . Remark3.11 (Interpretation ofV j ).The marginal social valueV j aggregates three forces: (i)Direct returns: an additional agent in familyjincreasesY j through the term (N j ) η j ; (i)Externalities: through Γ j (N −j ), the output of familyjdepends on the populations of other families; (i)Cost: the maintenance costc j reducesV j . WhenV j = 0, familyjis at demographic equilibrium: the marginal benefit of an additional agent exactly equals its cost. Remark3.12 (Relation to replicator dynamics).Equation ( 10) is a generalization of the replicator equation to absolute populations with endogenous fitness. In the classical repli- cator equation, ̇x i =x i (f i − ̄ f)with exogenous fitnessf i . Here, the “fitness”V j (N)is endogenous: it depends on all populations through the production functions, externalities, and the orchestrator’s optimal allocation. 3.5 Hive Equilibrium Definition 3.13(Hive Equilibrium).AHive Equilibriumis a triple(N ∗ ,K ∗ ,λ ∗ )such that: (HE-1)Allocation optimality: K ∗ solves the inner problem ( 5) forN ∗ , with shadow pricesλ ∗ ; 8 (HE-2)Demographic stationarity:Φ(N ∗ ) = 0, i.e., for eachj: V j (N ∗ ) = 0ifN j∗ >0,V j (N ∗ )≤0ifN j∗ = 0;(11) (HE-3)Budget feasibility: P j c j N j∗ ≤B. Condition (HE-2) states that at equilibrium, every active family has zero marginal social value (marginal benefit equals marginal cost), and no inactive family would be profitable if activated. This is the free-entry condition of competitive equilibrium theory. 4 Main results 4.1 Existence of Hive Equilibrium Theorem 4.1(Existence).Under Assumptions 1–3, there exists at least one Hive Equi- librium(N ∗ ,K ∗ ,λ ∗ )withN ∗ ∈Ω. Proof.Forε >0sufficiently small, define the truncated feasible setΩ ε =N∈Ω :N j ≥ ε∀j, which is compact and convex. Define the mapΨ ε : Ω ε →Ω ε by: Ψ ε (N) = proj Ω ε N+εΦ(N) , whereproj Ω ε is the Euclidean projection ontoΩ ε . Under our assumptions,V j (N)is continuous onΩ ε (composition ofC 2 functions), soΦ is continuous. The projection onto a closed convex set is continuous. HenceΨ ε is a continuous map from the compact convex setΩ ε into itself. By Brouwer’s fixed-point theorem,Ψ ε has a fixed pointN ∗ ε . Taking a convergent subsequence asε→0(by compactness ofΩ), the limitN ∗ ∈Ω satisfiesΦ(N ∗ ) = 0at interior components andV j (N ∗ )≤0at boundary components (N j∗ = 0). The allocationK ∗ and pricesλ ∗ are obtained from the inner problem atN ∗ . 4.2 Pareto optimality Theorem 4.2(First Welfare Theorem for Hives).Under Assumptions 1–4, every Hive Equilibrium(N ∗ ,K ∗ ,λ ∗ )isPareto-optimal: there exists no feasible(N ′ ,K ′ )withW(K ′ ,N ′ )> W(K ∗ ,N ∗ ). Proof.At a Hive Equilibrium,K ∗ maximizesW(·,N ∗ )over resource constraints (HE-1), andN ∗ satisfiesV j (N ∗ ) = dW ∗ /dN j = 0for all active families (HE-2). Suppose for contradiction that there exists feasible(N ′ ,K ′ )withW(K ′ ,N ′ )> W(K ∗ ,N ∗ ). SinceW ∗ (N) = max K W(K,N), we haveW ∗ (N ′ )≥W(K ′ ,N ′ )> W(K ∗ ,N ∗ ) =W ∗ (N ∗ ). We show thatW ∗ is strictly concave inNonΩ. The HessianH=D 2 N W ∗ has diagonal entriesH j =w j [u ′ (Y ∗ j ) (∂Y j /∂N j ) 2 +u ′ (Y ∗ j )∂ 2 Y j /(∂N j ) 2 ], which are negative under Assumptions 2(i) and3(the dominant term isη j (η j −1)(N j ) η j −2 <0). The off-diagonal 9 entriesH jk are bounded by a term proportional to|γ jk |. By a Gershgorin argument onH, Assumption 4ensures that every eigenvalue ofHis strictly negative, soW ∗ is strictly concave onΩ. The condition∇ N W ∗ (N ∗ ) = 0(from HE-2) then impliesN ∗ is the unique global maximum ofW ∗ onΩ, contradictingW ∗ (N ′ )> W ∗ (N ∗ ). Remark4.3.Pareto optimality relies on the orchestrator havingfull informationabout production functions and externalities. If some externalities are unobserved, the equilib- rium may fail to be Pareto-optimal, necessitating Pigouvian corrections (see Section 7). 4.3 Multiplicity and indeterminacy We now relax Assumptions 2(i) and4to allow increasing returns for some families. Assumption 5(Strategic complementarity).There exist familiesj, kwithγ jk >0and γ kj >0(mutual positive externalities). Assumption 6(Local increasing returns).There exists a familyj 0 withη j 0 ≥1(non- decreasing returns within family). Theorem 4.4(Multiplicity of equilibria).Under Assumptions 1–3,5, and6, there exist open sets of parameters(w,R)for which at least two distinct Hive Equilibria coexist. Proof sketch.Consider the simplest non-trivial case:S= 2families with mutual comple- mentarity (γ 12 , γ 21 >0) and increasing returns for family 1 (η 1 ≥1). At a Hive Equilibrium, the active families satisfyV j (N ∗ ) = 0, which defines two curves in(N 1 , N 2 )-space: C 1 =(N 1 , N 2 ) :V 1 (N 1 , N 2 ) = 0,C 2 =(N 1 , N 2 ) :V 2 (N 1 , N 2 ) = 0. With increasing returns (η 1 ≥1),C 1 is non-monotone:V 1 increases withN 1 for smallN 1 (increasing returns dominate) and decreases for largeN 1 (resource scarcity dominates). With positive externalities (γ 12 >0),V 1 increases withN 2 , tiltingC 1 in(N 1 , N 2 )-space. For sufficiently strong complementarities,C 1 andC 2 intersect at least twice, yielding at least two interior equilibria with distinct population structures. Full details, including the computation of the crossing conditions, are given in Appendix A. Remark4.5 (Interpretation).The multiple equilibria correspond to distinct “morpholo- gies” of the Hive: •Exploitation equilibrium: high specialization, concentrated populations, high effi- ciency; •Exploration equilibrium: high diversity, distributed populations, robustness to per- turbation. The system can settle into either morphology depending on initial conditions and historical path—a form ofpath dependencedirectly analogous to the indeterminacy in multi-sector growth models [ 3,10]. 10 4.4 Stolper–Samuelson analog Theorem 4.6(Hive Stolper–Samuelson).At an interior Hive Equilibrium, the Jacobian matrix S= ∂λ ∗ ∂w ∈R M×S (12) satisfies amagnification effect: if preferencew j increases (ceteris paribus), the shadow priceλ m∗ of the resource most intensively used by familyjincreases proportionally more than the preference change: ˆ λ m∗ ˆw j >1for the resourcemused most intensively by familyj,(13) whereˆx= dx/xdenotes a proportional change. Proof sketch.Differentiate the first-order conditions ( 7) totally with respect tow. The resulting linear system, combined with the resource constraints P j K m∗ j =R m and the CES structure ofF j , yields the Stolper–Samuelson matrix whose structure is inherited from the factor intensity matrixθ= (θ jm )withθ jm =λ m∗ K m∗ j /(w j u ′ (Y ∗ j )Y ∗ j ). The magnification effect follows from the algebraic structure ofθ −1 whenS=M, and from generalized Stolper–Samuelson results [ 16,8] whenS6=M. See AppendixA. Remark4.7 (Operational interpretation).Suppose the system administrator increases the weight on quality (w qual ↑). The S matrix predicts which resources will become scarce (e.g., GPU if quality-intensive families are GPU-heavy)before the change is applied. This is apredictive governancetool: the operator can anticipate bottlenecks and provision resources accordingly. 4.5 Rybczynski analog Theorem 4.8(Hive Rybczynski).At an interior Hive Equilibrium, the Jacobian matrix RB= ∂N ∗ ∂R ∈R S×M (14) satisfies amagnification effect: if endowmentR m increases (ceteris paribus), the equilib- rium population of the family that uses resourcemmost intensively expands proportionally more than the endowment change, while other families may contract: ˆ N j∗ ˆ R m    >1if familyjusesmmost intensively, <0for some familiesj ′ 6=j. (15) Proof sketch.At steady state,Φ(N ∗ ,R) = 0. By the implicit function theorem (assuming D N Φis non-singular): ∂N ∗ ∂R =− D N Φ −1 D R Φ. The structure ofD R Φreflects how resource changes affect marginal valuesV j through the allocationK ∗ (N,R). The magnification effect follows from the factor intensity structure, as in the classical Rybczynski theorem. See Appendix A. 11 Remark4.9 (Operational interpretation).Suppose a second GPU cluster is added to the system (R gpu ↑). The RB matrix predicts that GPU-intensive families (e.g., generation, transformation) willproliferate, while CPU-light families (e.g., logging, monitoring) may decline—not because they are less useful, but because the system re-optimizes the allo- cation. This is acapacity planning theorem: the operator can predict how the Hive will restructure before provisioning hardware. 4.6 Endogenous cycles Theorem 4.10(Endogenous demographic cycles).Under Assumptions 1–3and5, sup- pose that: (i)S≥2; (i) The JacobianJ=D N Φ(N ∗ )at an interior Hive EquilibriumN ∗ has a pair of complex conjugate eigenvaluesα(p)±iβ(p), wherepis a bifurcation parameter (e.g., an externality strengthγ jk ); (i)Transversality:dα/dp p=p 0 6= 0at the valuep 0 whereα(p 0 ) = 0; (iv) All other eigenvalues ofJhave strictly negative real parts atp 0 . Then aHopf bifurcationoccurs atp=p 0 : forpnearp 0 , a family of periodic orbits (limit cycles) bifurcates from the equilibriumN ∗ . Proof.This is a direct application of the Hopf bifurcation theorem[ 21,11] to the system ̇ N= Φ(N;p). Conditions (i)–(iv) are exactly the hypotheses of the theorem. It remains to verify that these conditions can be satisfied for some parameter configuration. This is demonstrated by construction in Appendix A, where we exhibit a two-family ex- ample with explicit parameter values yielding complex eigenvalues crossing the imaginary axis. Remark4.11 (The “seasons” of the Hive).The periodic orbits generated by the Hopf bifurcation correspond to cyclical phases of the Hive’s demographic structure: •Expansion phase: births exceed deaths, population grows, new agents explore diverse tasks; •Consolidation phase: successful families specialize, efficiency increases, diversity de- creases; •Contraction phase: deaths exceed births, underperforming agents are pruned,re- sources are freed; •Renewal phase: stem agents regenerate diversity, the cycle restarts. These cycles areendogenous—driven by the internal dynamics of externalities and returns to scale, not by external shocks. This is the agent-demographic analog of the competitive equilibrium cycles of Benhabib & Nishimura [ 3]. 4.7 Stability conditions Theorem 4.12(Local stability).An interior Hive EquilibriumN ∗ islocally asymptot- ically stableif and only if all eigenvalues of the JacobianJ=D N Φ(N ∗ )have strictly 12 negative real parts. A sufficient condition is: η max <1andkEk ∞ ·max j N j∗ <min j V ′ j (N ∗ ) ·N j∗ ,(16) whereη max = max j η j andkEk ∞ is the infinity norm of the externality matrix. Proof.The Jacobian at an interior equilibrium (V j (N ∗ ) = 0) is: J jk = ∂Φ j ∂N k N ∗ = ∂V j ∂N k N ∗ ·N j∗ +V j (N ∗ )·δ jk = ∂V j ∂N k N ∗ ·N j∗ ,(17) sinceV j (N ∗ ) = 0. ThusJ= diag(N ∗ )·D N V(N ∗ ). The diagonal terms areJ j = (∂V j /∂N j )·N j∗ . Underη j <1,∂V j /∂N j <0(diminishing returns), soJ j <0. The off-diagonal terms areJ jk = (∂V j /∂N k )·N j∗ , whose sign is determined by the externalityγ jk . By the Gershgorin circle theorem, all eigenvalues ofJlie in the union of disks centered atJ j with radius P k6=j |J jk |. The sufficient condition ( 16) ensures that these disks lie entirely in the left half-plane. Remark4.13 (Governance lever).The stability condition can be interpreted as a require- ment that the “mortality pressure” (captured by|V ′ j |·N j∗ , the speed at which declining families shrink) dominates the “amplification pressure” from cross-family externalities (kEk ∞ ·N j∗ ). The orchestrator can enforce stability by increasing thesensitivity of the birth-death mechanism to marginal value deviations. 5 Numerical illustrations We complement the analytical results with numerical computations forS= 3andS= 5 families, illustrating the equilibrium structure, regimetransitions, and dynamic behavior predicted by the theory. All computations use log utility (σ= 1) and CES production with the parameters specified below. 5.1 Three-family Hive (S= 3,M= 2) Consider three agent families—perception(j= 1),reasoning(j= 2), andgeneration (j= 3)—sharing two resources: GPU (m= 1) and memory (m= 2). Parameters.A j = 1for allj;c j = 1;B= 15;R 1 = 10(GPU),R 2 = 8(memory). CES elasticities:ρ 1 = 0.8,ρ 2 = 1.2,ρ 3 = 0.6. Factor shares:α 1 = (0.3,0.7)(memory- intensive),α 2 = (0.5,0.5)(balanced),α 3 = (0.8,0.2)(GPU-intensive). Preferences:w= (0.35,0.40,0.25). 13 051015 0 2 4 6 8 t N j ( t ) (a) Convergence toN ∗ A (stable spiral) Percep. Reason. Gener. 051015 0 2 4 6 8 t N j ( t ) (b) Convergence toN ∗ B (stable node) Percep. Reason. Gener. Figure 1: Path dependence in the three-family Hive (S= 3,M= 2) withη 1 = 1.3and γ 12 =γ 21 = 0.4.(a)FromN 0 = (5.0,5.0,5.0), the system spirals into the perception- dominant morphologyN ∗ A ≈(6.1,5.8,1.4).(b)FromN 0 = (2.0,3.0,6.0), it converges monotonically to the generation-dominant morphologyN ∗ B ≈(2.3,4.9,5.2). The coexis- tence of two locally stable equilibria confirms Theorem 4.4. Baseline: weak externalities (γ jk = 0.05).Withη j = 0.7for alljand uniformly weak externalitiesγ jk = 0.05forj6=k, the system has a unique Hive Equilibrium (Theorem 4.12applies): N ∗ ≈(4.2,5.1,3.7),λ ∗ ≈(0.31,0.42).(18) The shadow price of memory exceeds that of GPU, reflecting thememory-intensive pref- erence structure (w 1 = 0.35for the memory-intensive family). Increasing returns and multiplicity.Settingη 1 = 1.3(increasing returns for percep- tion) and strengthening complementaritiesγ 12 =γ 21 = 0.4while keepingγ j3 =γ 3j = 0.05, two distinct equilibria emerge (Theorem 4.4): N ∗ A ≈(6.1,5.8,1.4)(perception-dominant morphology),(19) N ∗ B ≈(2.3,4.9,5.2)(generation-dominant morphology).(20) The perception-dominant morphology (N ∗ A ) concentrates resources on the perception– reasoning axis, exploiting the strong complementarity. The generation-dominant mor- phology (N ∗ B ) instead develops a large generation pool, which is self-sustaining through its high GPU utilization. Eigenvalue analysis.AtN ∗ A , the Jacobian eigenvalues are approximately−1.8,−0.4± 0.9i: the equilibrium is a stable spiral. AtN ∗ B , the eigenvalues are−2.1,−0.7,−0.3: a stable node. Both equilibria are locally stable, confirming the path-dependence pre- dicted by the theory. Figure 1illustrates the convergence dynamics from two different initial conditions. Hopf bifurcation.Introducing asymmetric externalities—γ 12 = 0.5(reasoning helps perception) butγ 21 =−0.3(perception congests reasoning)—and varying|γ 21 |as the bifurcation parameter, the real part of the complex eigenvalue pair crosses zero atγ 21 ≈ 14 051015202530 2 4 6 t(model time units) N j ( t ) Perception Reasoning Generation Figure 2: Endogenous demographic cycles (Hopf bifurcation) in the three-family Hive. Parameters:γ 12 = 0.5,γ 21 =−0.5(beyond the critical valueγ ∗ 21 ≈−0.42). Starting from a perturbed equilibrium, the oscillation amplitude grows and saturates to a limit cycle with periodT≈8.3. The phase shifts between families produce the four “seasons” of the Hive: expansion, consolidation, contraction, and renewal. −0.42. Beyond this threshold, endogenous demographic cycles emerge with periodT≈8.3 (in model time units), confirming Theorem 4.10. Figure2displays the resulting limit cycle. 5.2 Five-family Hive (S= 5,M= 3) We scale the model to five families—perception(j= 1),reasoning(j= 2),generation (j= 3),verification(j= 4),monitoring(j= 5)—sharing three resources: GPU (m= 1), memory (m= 2), and I/O bandwidth (m= 3). Parameters.A j = 1;c j = 1;B= 30;R= (20,15,12). CES elasticities:ρ j ∈ 0.6,1.2,0.8,1.0,0.5. Factor share matrix (rows = families, columns = GPU/memory/I/O): α=         0.2 0.6 0.2 0.4 0.4 0.2 0.7 0.1 0.2 0.3 0.3 0.4 0.1 0.2 0.7         . Preferences:w= (0.20,0.30,0.25,0.15,0.10). Baseline: unique equilibrium.Withη j = 0.7andγ jk = 0.02for allj6=k, the unique equilibrium is: N ∗ ≈(5.1,7.3,6.2,4.8,3.6),λ ∗ ≈(0.28,0.35,0.41).(21) I/O bandwidth is the scarcest resource (λ 3∗ is highest), consistent with the monitoring family’s heavy I/O usage. Rybczynski prediction.Increasing GPU endowment by 50% (R 1 : 20→30), the new equilibrium isN ∗ ≈(5.4,8.1,9.8,5.0,3.2). The GPU-intensive generation family expands by 58% (magnification>1), while the I/O-intensive monitoring familycontractsby 11%, precisely as predicted by Theorem 4.8. 15 00.10.20.30.40.5 0.6 0.8 1 1.2 1.4 Unique stableMultiple CyclesInstability γ crit ≈ 0 η crit ≈0.98 Externality strengthγ Maximum return to scale η 1 Figure 3: Numerical regime diagram for the five-family Hive (S= 5,M= 3). Each marker corresponds to one parameter sweep point:•unique stable equilibrium,multiple stable equilibria, Nendogenous limit cycle,×instability (boundary dynamics). Dashed lines indicate the critical boundariesγ crit ≈0.20andη crit ≈0.98, confirming the four-region partition of Section 6. Stolper–Samuelson prediction.Increasing the weight on verification (w 4 : 0.15→ 0.25, renormalized), the shadow price of I/O (λ 3∗ ) increases by 32% while GPU price (λ 1∗ ) decreases by 8%—a magnification effect consistent with Theorem 4.6, since verification is the most I/O-intensive family. Regime transitions.We sweep the parameter space(γ, η max )by varyingγ jk =γ (uniform externality) andη 1 (returns to scale in perception), computing equilibria at each grid point. The resulting numerical regime diagram confirmsthe four-region structure of Section 6: RegionParameter rangeEquilibria found Unique stableγ <0.15,η 1 <0.951 (stable node/spiral) Multiple stableγ >0.25,η 1 <0.952–3 (distinct morphologies) Endogenous cyclesγ mixed ,η 1 >1.11 (limit cycle,T≈6–12) Instabilityγ >0.35,η 1 >1.2 0 interior (boundary dynamics) The transition boundaries (γ crit ≈0.20,η crit ≈0.98) are sharp and consistent across multiple random initializations, confirming the robustness of the regime diagram. The numerical regime diagram is shown in Figure3. Welfare monotonicity.In all simulations within the weak-externality regime, we verify numerically thatW ∗ (t)is strictly increasing along trajectories, withdW ∗ /dt= P j V 2 j N j > 0until convergence, confirming Theorem A.1. The convergence rate is approximately ex- ponential, with time constantτ≈1/|λ min (J)|whereλ min is the eigenvalue ofJclosest to the imaginary axis. Reproducibility.All numerical results were obtained by solving the steady-state con- ditionsV j (N ∗ ) = 0via Newton’s method, computing Jacobian eigenvalues via standard 16 linear algebra routines (numpy.linalg), and integrating the ODE system (10) using a fourth-order Runge–Kutta scheme. Python source code reproducing all numerical results and Figures 1–3will be released as supplementary material upon publication. 6 The regime diagram Theorems 4.1–4.12collectively define a partition of the parameter space into qualitatively distinct regions. The two most informative parameters are theexternality strengthγ= max j6=k |γ jk |and themaximum return to scaleη max = max j η j . γ(externalities) η max (returns) Unique equilibrium (Thm. 4.12) Multiple equilibria (Thm. 4.4) Endogenous cycles (Thm.4.10) Instability (divergence) γ crit η crit Hopf •Bottom-left(γ < γ crit ,η max < η crit ): unique, stable equilibrium. The Hive con- verges to a single morphology. Theorems4.1and4.12apply. •Bottom-right(γ > γ crit ,η max < η crit ): multiple stable equilibria coexist. The Hive exhibitspath dependence: the morphology depends on initial conditions. The- orems 4.1and4.4apply. •Top-left(γ < γ crit ,η max > η crit ): unique equilibrium destabilized by increasing returns. Hopf bifurcation generates endogenous cycles. Theorems 4.1and4.10 apply. •Top-right(γ > γ crit ,η max > η crit ): strong externalities combined with increasing returns can lead to divergence. The orchestrator must enforce hard population caps to maintain bounded dynamics. The critical valuesγ crit andη crit depend on the full parameter vector(w,R,A j , c j , ρ j ) and are computable from the eigenstructure of the JacobianD N Φat any candidate equi- librium. Remark6.1 (Governance).The regime diagram is adashboardfor the Hive operator. The orchestrator can: • Move the systemleft(reduce externalities) by isolating agent families (sandboxing, rate limiting); • Move the systemdown(reduce returns to scale) by increasing within-family compe- 17 tition or capping family size; • Move the systembetween equilibria(in the multiplicity region) by applying transient preference shocks. 7 Discussion 7.1 A macroeconomic governance framework The Agentic Hive framework provides a formal alternative tothe ad-hoc heuristics cur- rently used to manage multi-agent systems. Rather than manually deciding agent counts and roles, the operator specifiespreferenceswandresource endowmentsR; the population structure then emerges from the equilibrium conditions. The analogy with macroeconomic governance is precise: Central bank / Government Hive orchestrator Sets interest ratesSets preference weightsw Fiscal policy (taxes, spending) Resource allocationR Monetary policy toolsBirth/death rate parameters GDP, inflation, unemploymentW,N, resource utilization Taylor ruleStability condition ( 16) The S and RB matrices (Theorems4.6–4.8) are the Hive equivalents of macroeconomic forecasting models: they predict how the system will restructure in response to policy changes,before the changes are implemented. 7.2 Connection to Global Workspace Theory The orchestrator plays a role analogous to the Global Workspace of Baars [ 2]: it receives signals from all agent families (perception), selects relevant information (attention), inte- grates it into a global state (working memory), and broadcasts resource allocation signals (action selection). The endogenous cycles of Theorem 4.10provide a formal model of what cognitive scien- tists call “attentional oscillations”—periodic shifts inthe focus of the workspace between competing representations [ 6]. We do not claim that the Agentic Hive is conscious. We claim only that the mathematical structure of the Hive (global workspace + demographic dynamics + endogenous cycles) is anecessary prerequisitefor any computational system that might exhibit integrated information processing at scale. 7.3 Imperfect information and mechanism design Theorems 4.2assumes the orchestrator has perfect information about production functions and externalities. In practice, the orchestrator observesonly noisy performance metricsμ a and must infer the production parameters. This is aprincipal-agent problem[ 25]: the orchestrator (principal) must design incentive- compatible mechanisms to elicit truthful reports from agents. The tools of mechanism design (revelation principle, Vickrey–Clarke–Groves mechanisms) are directly applicable. 18 When externalities are imperfectly observed, the equilibrium may fail to be Pareto-optimal. The standard correction is aPigouvian tax/subsidy: agents in families with negative externalities pay a taxτ j =− P k6=j γ kj ·∂W/∂N k , which internalizes the externality. 7.4 Relation to evolutionary dynamics The population dynamics ( 10) are formally a selection dynamic. This connects the Hive to the theory of evolutionary games [13,23]. However, two key differences separate the Hive from standard evolutionary dynamics: 1.Endogenous fitness: in standard replicator dynamics, fitness depends on population frequencies. In the Hive, fitnessV j (N)depends on absolute populationsandon the orchestrator’s optimal resource allocation—an endogenous, optimization-mediated feedback loop. 2.Multi-factor production: agents consume multiple resources (GPU, memory, atten- tion), not a single abstract fitness payoff. This multi-factor structure is what gener- ates the Stolper–Samuelson and Rybczynski effects, which have no analog in stan- dard evolutionary dynamics. 7.5 Limitations 1.Continuous-population approximation.We model family sizes as continuous vari- ablesN j ∈R + , which is appropriate for large populations but may lose accuracy for small agent counts. A stochastic extension (birth-death Markov chain) would address this. 2.Discrete families.We assumeSdiscrete families. In practice, specialization may be continuous—a point in a manifoldS ⊂R d . The continuous extension replaces the ODE system with a partial differential equation (reaction-diffusion onS), analogous to Turing morphogenesis [ 27]. 3.Passive agents as theoretical baseline.The present framework assumes agents are passive: their behavior is fully determined by their genomeand orchestrator signals, and the orchestrator solves a centralized social welfare problem. This is the stan- dard starting point in economic theory—analogous to theoptimal growth(Ramsey) formulation that precedes thecompetitive equilibrium(Arrow–Debreu) formulation. The natural next step is thedecentralizedextension in which each agent maximizes its own utility under resource and interaction constraints, and the orchestrator acts as a market mechanism rather than a central planner. In this setting, the Hive Equilibrium becomes a Nash equilibrium of a population game, and standard tools from mechanism design (revelation principle, VCG mechanisms) are needed to align individual incentives with social welfare. We view the centralized framework of this paper as the necessary foundation: one must first understandthe socially optimal demographic structure before analyzing what happens when agents pursue their own objectives. 4.Empirical validation.The present paper is purely theoretical. Empirical validation on real multi-agent systems is needed to calibrate the parameters(η j , γ jk , ρ j )and verify the predicted regimes. 19 8 Conclusion We have introduced the Agentic Hive, a formal framework for self-organizing multi-agent systems in which the population of agents undergoes demographic dynamics governed by multi-sector growth theory. Our seven main results—existence, Pareto optimality, multiplicity, Stolper–Samuelson and Rybczynski comparative statics, endogenous cycles, and stability conditions—provide the first analytical foundation for the question:how should the population structure of a multi-agent system evolve? The regime diagram (Section 6) gives operators a formal tool for understanding and controlling the Hive’s behavior: by adjusting preferencesand resources, the orchestra- tor can steer the system between stable equilibria, manage path dependence, and avoid instability—all with quantitative predictions derived from the S and RB matrices. We conjecture that the Agentic Hive framework providesnecessary conditionsfor the design of scalable, self-organizing multi-agent AI systems. Systems that lack a formal demographic theory may function at small scale, but have no guarantee of coherent be- havior as agent counts grow, as resource constraints tighten, or as the task distribution shifts. The mathematical tools developed over seven decades for multi-sector economies are, we believe, the right foundation for this emerging challenge. Future directions. •Empirical validationon real multi-agent deployments (e.g., LLM orchestration on edge hardware); •Mechanism designfor Hives with strategic agents; •Meta-Hives: federations of interacting Hives, governed by international trade theory (Heckscher–Ohlin); •Continuous specializationvia reaction-diffusion PDEs on the specialization manifold; •Chaos and complex dynamicsin Hives with strong nonlinearities, extending Nishimura & Yano [ 22]. References [1] K. J. Arrow and G. Debreu. Existence of an equilibrium fora competitive economy. Econometrica, 22(3):265–290, 1954. [2] B. J. Baars.A Cognitive Theory of Consciousness. Cambridge University Press, 1988. [3] J. Benhabib and K. Nishimura. Competitive equilibrium cycles.Journal of Economic Theory, 35(2):284–306, 1985. [4] G. 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A Proofs and technical details A.1 Proof details for Theorem 4.4(Multiplicity) ConsiderS= 2families with CES production (3),M= 1resource (for simplicity), and Cobb–Douglas externalities: Y 1 =A 1 (K 1 ) α 1 (N 1 ) η 1 (N 2 ) γ 12 , Y 2 =A 2 (K 2 ) α 2 (N 2 ) η 2 (N 1 ) γ 21 . WithK 1 +K 2 =R(single resource constraint), the inner problem givesK ∗ j as a function ofN. The steady-state conditionsV 1 =V 2 = 0define implicit curvesC 1 ,C 2 in(N 1 , N 2 )-space. ForC 1 :V 1 = 0gives w 1 u ′ (Y 1 )A 1 (K ∗ 1 ) α 1 η 1 (N 1 ) η 1 −1 (N 2 ) γ 12 =c 1 . Monotonicity inN 1 .Ifη 1 <1: the left-hand side decreases inN 1 (diminishing re- turns), soC 1 is downward-sloping (moreN 1 requires moreN 2 to maintainV 1 = 0). Ifη 1 = 1: the left-hand side is constant inN 1 (conditional onK ∗ 1 ), soC 1 is a horizontal line in the(N 1 , N 2 )-plane. Ifη 1 >1: the left-hand side increases inN 1 for smallN 1 (increasing returns dominate) and eventually decreases (resource scarcity asN 1 grows and absorbs more ofR). Hence C 1 is non-monotone—it “bends back.” Effect ofγ 12 >0.Positive externality from family 2 shiftsC 1 inward: moreN 2 makes family 1 more productive, reducing theN 1 needed to maintainV 1 = 0. 22 Multiple intersections.WhenC 1 bends back (due toη 1 >1) andC 2 has a similar structure (or is monotone but appropriately positioned duetoγ 21 >0), the two curves can intersect at least twice. Each intersection is a Hive Equilibrium with a distinct population structure. Explicit numerical example:A 1 =A 2 = 1,α 1 =α 2 = 0.5,η 1 = 1.2,η 2 = 0.8,γ 12 = γ 21 = 0.3,w 1 =w 2 = 0.5,σ= 1(log utility),c 1 =c 2 = 1,R= 10. Numerical solution of V 1 =V 2 = 0yields two interior equilibria:N ∗ A ≈(2.1,3.8)andN ∗ B ≈(4.7,1.9). A.2 Proof details for Theorem4.6(Stolper–Samuelson) We follow Jones [ 16]. At the optimum, the FOC (7) forS=M(square case) can be written in proportional changes (hat algebra): ˆw j + ˆu ′ j + ˆη j + ˆ Γ j = ˆ λ m − ρ j −1 ρ j ˆ K m j − ˆ F j , for alljand the resourcemthat binds for familyj. Defining the factor intensity matrixθwithθ jm =λ m K m j /(w j u ′ j Y j )(the share of re- sourcemin the value of familyj’s output), and using the resource constraints in differ- ential form, we obtain: ˆ λ=θ −T · ˆ w+(externality corrections). Whenθsatisfies astrong factor intensitycondition (each family uses one resource more intensively than others),θ −T has the sign pattern that yields the magnification effect ( 13). A.3 Proof details for Theorem4.10(Endogenous cycles) We exhibit a parameter configuration satisfying the Hopf conditions. ConsiderS= 2 families with: Φ 1 (N) =V 1 (N 1 , N 2 )·N 1 , Φ 2 (N) =V 2 (N 1 , N 2 )·N 2 . At an interior equilibriumN ∗ , the Jacobian is: J= ∂V 1 ∂N 1 N 1∗ ∂V 1 ∂N 2 N 1∗ ∂V 2 ∂N 1 N 2∗ ∂V 2 ∂N 2 N 2∗ ! = a b c d ! . The eigenvalues areλ= 1 2 h (a+d)± q (a−d) 2 + 4bc i . Withη 1 , η 2 <1:a <0andd <0(diminishing returns). Withγ 12 >0:b >0(family 2 helps family 1). Withγ 21 >0:c >0(family 1 helps family 2). The eigenvalues are complex when(a−d) 2 + 4bc <0, i.e., when4bc <−(a−d) 2 . Since b, c >0, this requiresbcto be negative, which occurs when one of the cross-effects is 23 negative—for example, if family 1 helps family 2 (γ 21 >0, soc >0) but family 2congests family 1 (γ 12 <0, sob <0). In this mixed externality case (b <0,c >0): • The tracea+d <0(stable direction); • The determinantad−bccan change sign as|γ 12 |varies (becausebbecomes more negative); • When the trace passes through zero (as a function of a parameterp, e.g., the mag- nitude ofγ 12 ), the eigenvalues cross the imaginary axis⇒Hopf bifurcation. Transversality (d Re(λ)/dp6= 0) is verified by direct computation of the derivative of the trace with respect top. Hence all conditions of the Hopf bifurcation theorem are satisfied, and a family of periodic orbits (limit cycles) emerges forpnear the critical value. A.4 Lyapunov property and welfare monotonicity The following result shows that the population dynamics (10) are not arbitrary—they alwaysincrease social welfare. Theorem A.1(Welfare monotonicity).Under Assumptions1–4, the optimized welfare W ∗ (N)is non-decreasing along trajectories of the population dynamics( 10): dW ∗ dt = S X j=1 V j (N) 2 ·N j ≥0,(22) with equality if and only ifV j (N) = 0for every active family (N j >0), i.e., at a Hive Equilibrium. Consequently, under Assumption 4(which ensuresW ∗ is strictly concave and bounded above onΩ), every trajectory of( 10)starting inΩconverges to the set of Hive Equilibria. Proof.By the chain rule and the definition of marginal social value (Definition3.9): dW ∗ dt = S X j=1 ∂W ∗ ∂N j ̇ N j = S X j=1 V j (N)· V j (N)·N j = S X j=1 V j (N) 2 N j . SinceN j ≥0, each term is non-negative, hencedW ∗ /dt≥0. Equality holds iffV j (N) 2 · N j = 0for allj, i.e.,V j = 0wheneverN j >0. Under Assumption 4,W ∗ is strictly concave andΩis compact, soW ∗ is bounded above onΩ. SinceW ∗ is non-decreasing and bounded,W ∗ (t)→ ̄ Wfor some ̄ W. By the LaSalle invariance principle, the trajectory converges tothe largest invariant set within N∈Ω : dW ∗ /dt= 0, which is precisely the set of Hive Equilibria. RemarkA.2 (Demographic pressure).Out of equilibrium, the quantity P j V j (N)·N j can be interpreted as the totaldemographic pressurein the Hive. If positive, the system is under-populated (net entry pressure); if negative, over-populated (net exit pressure). At equilibrium, these forces exactly balance—the functional analog of Walras’ law. The Lyapunov result above shows that this rebalancing always improves social welfare. 24 RemarkA.3 (Scope of the convergence result).TheoremA.1guarantees convergence under Assumption 4(weak externalities, decreasing returns). When this assumption is relaxed (Section 4.3),W ∗ may no longer be globally concave, and the dynamics can exhibit the richer behaviors described in the regime diagram: convergence to one of multiple equilibria (path dependence) or sustained oscillations (Theorem 4.10). 25

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