{"id":"b0636567-6975-45c8-bda6-917dfd1ad59a","arxiv_id":"2502.03872","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two sub-populations sharing one resource pool, an equilibrium ratio exists only when their reproduction-weighted claim survival probabilities at the common threshold are equal and at least 1.","lead":"This paper proves the mathematical conditions for two populations that share a common resource pool to reach a stable size ratio. The key equation connects each population's reproduction rate, resource production, and claim-size distribution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Part (IV) proves only marginal survival of the home process; the step to joint survival with a finite ratio uses the limit α whose existence is at stake.","rationale":"Reader's verdict is CONDITIONAL and I agree. The strongest claim has a necessary-direction proof that is broadly plausible, but the sufficiency direction has a genuine logical gap. Part (IV) attempts to prove positive probability of joint survival of both sub-processes and convergence of their ratio, yet it reduces this to marginal survival of the home process via the claim that 'if at least one ... tends to infinity with strictly positive probability, then both must do so'. That inference is unjustified because it assumes the existence of the a.s. limit α, exactly what must be established. This is an internal step, not a matter of external consensus. The reader's rationale already notes that sufficiency is delegated to a survival theorem, so my agreement is partial: the reader's weakest_assumption was the additive resource-space model, which is an explicit modeling assumption and not the load-bearing flaw. Other issues, such as compressed Borel-Cantelli arguments in Part (III) and the probability/expectation notation around (9), are real but secondary. If the sufficiency gap is repaired, those may be manageable. I would keep the verdict CONDITIONAL: the necessary part and equilibrium equation appear correct under the stated assumptions, but the paper should not be accepted as a complete proof until Part (IV) supplies a valid argument for joint survival and ratio convergence.","tokens_in":10452,"tokens_out":10122,"duration_ms":115107,"concrete_test":"Re-derive Part (IV) for the model with point-mass claim distributions F_h=F_i=δ_1 and r_h=r_i=2, R_h=R_i=1, and offspring laws with positive extinction probabilities. Compute the coupled two-type RDBP path probabilities for large t: specifically P(Γ^h_t→∞, Γ^i_t→∞) and P(Γ^h_t→∞, Γ^i_t→0). If the second event has positive probability, the inference from (12) to (11) is false; if it is zero, the missing coupling argument should be written out explicitly. Either outcome settles whether the sufficiency proof is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The sufficiency direction does not establish (11). Equation (12) shows P(Γ^h_t→∞ | r_hF_h(F)>1)>0 via the 2015 survival theorem. The next sentence claims that if one sub-process tends to infinity with positive probability then both must, 'according to the definition of α'. This is circular: the definition of α as the a.s. limiting ratio presupposes exactly the joint convergence that must be proved. On the positive-probability event Γ^h→∞ while Γ^i dies out, the ratio tends to 0, and nothing in Part (I) rules this out, since Part (I) applies only when a finite limit α already exists. The conditioning 'given that Γ^i_t/Γ^h_t ↛0 and ↛∞' at the start of Part (IV) is not present in (12), so it cannot supply the missing jointness. Thus the paper leaves unproved the claimed positive-probability existence of an equilibrium under the >1 condition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This supplement to the author's Festschrift paper (Bruss, 2024) provides the proof of Theorem 3 for a model of two interacting resource-dependent branching processes, one 'home' and one 'immigrant' population. The theorem characterizes equilibria via a limiting ratio alpha of population sizes, a common resource threshold F, and an equilibrium equation balancing expected resource consumption against expected resource production, together with a constraint qualification on the reproduction rates times the claim distributions at F. The proof has four parts: (I) necessity of both sub-processes tending to infinity if a finite positive limit alpha exists; (II) derivation of the equilibrium equation from the common-resource balance via the strong law of large numbers; (III) necessity of the constraint qualification r_h F_h(F)=r_i F_i(F) >= 1; and (IV) sufficiency of the strictly greater-than-1 condition for the existence of an alpha-equilibrium with strictly positive probability. The paper also contains brief discussion of optimal transport, control, and multi-population extensions.","tokens_in":10656,"tokens_out":9475,"duration_ms":97199,"significance":"If the theorem is correct, it gives a clean equilibrium condition for two interacting sub-populations sharing a common resource, with a transparent economic interpretation and potential applications to population dynamics and resource allocation. The proof strategy is largely elementary, combining strong-law-of-large-numbers arguments with Borel-Cantelli techniques, and Part (II) in particular is a clear and convincing derivation of the equilibrium equation from the assumed resource balance. The necessity direction (Parts I-III) appears sound modulo minor presentation issues. However, the sufficiency direction (Part IV) contains a serious logical gap: it attempts to prove positive-probability existence of an equilibrium by first proving marginal survival of the home process and then inferring joint survival from the 'definition' of alpha, which presupposes the very convergence that must be established. This gap is load-bearing for the sufficiency claim, and the manuscript as it stands does not prove the existence part of the theorem.","major_comments":[{"comment":"The sufficiency proof is circular. Equation (12) establishes only that the home process survives with positive probability, P(Gamma^h_t -> infinity | r_h F_h(F) > 1) > 0, by invoking Theorem 4.4 of Bruss-Duerinckx (2015). The next sentence claims that if one sub-process tends to infinity then both must do so 'according to the definition of alpha', but the definition of alpha as the a.s. limiting ratio Gamma^i_t/Gamma^h_t presupposes exactly the joint convergence that needs to be shown. Nothing in Part (I) rules out the positive-probability event Gamma^h_t -> infinity while Gamma^i_t -> 0, since Part (I) applies only when a finite limit alpha already exists. The conditioning event {Gamma^i_t/Gamma^h_t not-> 0} ∩ {Gamma^i_t/Gamma^h_t not-> infinity} mentioned at the start of Part (IV) does not appear in (12), so it cannot supply the missing jointness. Consequently, the claimed positive-probability existence of an alpha-equilibrium under the strict inequality is unproved.","section":"Part (IV), Eq. (11)-(12)"},{"comment":"The marginal home process is not a resource-dependent branching process in the sense of Bruss-Duerinckx (2015), because in the two-population coupled model the resource threshold F_t depends on the total claims and total resources of both sub-populations (see equation (4) and the sentence before it). Therefore Theorem 4.4 of Bruss-Duerinckx (2015) cannot be applied directly to the marginal process (Gamma^h_t) without an additional argument that the coupling does not substantially change its survival behavior. Such an argument is not provided. This is a second, independent reason why the sufficiency proof fails.","section":"Part (IV), line 'the process (Gamma^h_t) is a RDBP by definition'"},{"comment":"The proof that r_h F_h(F) >= 1 and r_i F_i(F) >= 1 are necessary for an equilibrium is sketched via a Borel-Cantelli type argument citing Bruss (1978), but the argument is not fully written out. In particular, the step from bounded conditional expectation to almost-sure extinction is stated without derivation. This is a weaker concern than the Part (IV) gap, but if the authors intend this supplement to be a complete proof, this step should be expanded.","section":"Part (III), necessity of >= 1"}],"minor_comments":[{"comment":"The text reads 'the conditions r_h F_h(F) >= 1 and r_h F_h(F) >= 1 are necessary'; the second condition should be r_i F_i(F) >= 1.","section":"Part (III), paragraph after Eq. (8)"},{"comment":"Equation numbering is confusing: the paper refers to equation (14) of the main paper but also introduces a numbered equation (14) in Section 0.6 for the optimal transport problem. Please renumber or distinguish the supplement's equations to avoid ambiguity.","section":"Throughout"},{"comment":"Notation for resource production means is inconsistent: the abstract uses R_h and R_i, while Part (II) uses F_h and F_i for both claim distributions and production means in the same formulas. Please introduce distinct, consistently used symbols.","section":"Notation"},{"comment":"The optimal transport and control sections are speculative and not clearly connected to the theorem proved in the earlier parts. They would be better placed in a discussion section or in the main Festschrift paper, and their current inclusion distracts from the proof.","section":"Sections 0.6-0.8"},{"comment":"The event in (11) is stated with conditioning only on r_h F_h(F)=r_i F_i(F)>1, but the introductory sentence of Part (IV) conditions on the ratio not tending to 0 or infinity; the relationship between these two statements should be clarified.","section":"Proof of Part (IV), Eq. (11)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a supplement to the author's own Festschrift paper and relies heavily on a chain of the author's prior results (Bruss 1978, 1980; Bruss-Duerinckx 2015; Bruss 2021). The necessity direction of the theorem appears sound, and the SLLN-based derivation in Part (II) is a genuine strength. However, the sufficiency proof in Part (IV) has a fundamental circularity and also misapplies the single-type RDBP survival theorem to a marginal process of a coupled system. These are not merely presentation issues; they leave the existence claim unproved. If the authors can supply a correct proof of joint survival with positive probability (perhaps via a genuine two-type analysis), the paper could be acceptable, but as it stands the sufficiency part is not established. I recommend major revision and would want the revised version to be re-evaluated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper delivers what it promises: complete proofs of the equilibrium theorem announced in the Festschrift article. Parts (I)-(III), the necessity direction, are mostly sound. The SLLN derivation of the equilibrium equation in Part (II) is clean, and the argument that an existing finite ratio forces both sub-processes to infinity is fine. The Borel-Cantelli step in Part (III) is compressed but plausible, and citing Bruss-Duerinckx (2015) for the survival criterion is legitimate when the cited theorem is proved there.\n\nThe soft spot is Part (IV), and it is not a minor compression. To prove sufficiency the paper needs a strictly positive probability that both sub-processes go to infinity. What (12) proves is only that the home process goes to infinity with positive probability, via the 2015 theorem. The next sentence claims that if one sub-process tends to infinity then both must, 'according to the definition of α'. That is circular: α is defined as the a.s. limiting ratio, so invoking it presupposes exactly the joint convergence at issue. On the event where the home process grows and the immigrant process dies out, the ratio tends to 0, and nothing in Part (I) rules that out, because Part (I) applies only when a finite positive limit already exists. The initial conditioning in Part (IV) on the ratio neither going to 0 nor ∞ is not used in (12), so it cannot supply the missing jointness. As written, the positive-probability existence of an equilibrium under the strict >1 condition is not established.\n\nThe reader's conditional verdict is fair, but I'd go a bit further: the gap is load-bearing, not stylistic. The additive resource-claims assumption is stated clearly and is genuinely necessary for the linear equation; that is a modeling constraint, not a flaw. Minor issues: the text says 'Part (VI)' where it means 'Part (IV)', and the theorem statement lives only in the Festschrift paper, which is awkward for a supplement. The optimal transport and control sections are programmatic, as the author acknowledges.\n\nWho gets value from this? Anyone working on resource-dependent branching processes or demographic coexistence. The necessity result is a useful criterion, and the sufficiency gap is instructive — it marks exactly where a two-type survival argument must go. This deserves a serious referee: the core idea is valuable and the flaw is identifiable and likely fixable with an additional argument. Send it to review, but the referee should focus on Part (IV).","headline":"The necessity half of the equilibrium theorem is honestly proved; the sufficiency half has a genuine gap that needs a serious referee, but the paper deserves one.","tokens_in":11151,"tokens_out":4232,"would_cite":true,"duration_ms":41433,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J85"],"pacs":[],"model":"deepseek-v4-flash","headline":"A stable population ratio forces a unique resource threshold.","keywords":["equilibrium equation","resource-dependent branching process","Borel-Cantelli lemma","bisexual branching process","multi-type process","optimal transport","BRS-inequality","control"],"falsifier":"Construct a two-population resource-dependent branching process satisfying all assumptions except that the two sub-populations draw from separate resource spaces; if a finite positive limiting ratio $\\alpha$ still exists, the claimed equilibrium equation need not hold. Alternatively, simulate the model with claim densities such that $r_h F_h(F) = r_i F_i(F) = 1$ and check whether the ratio never converges to a finite positive constant, contradicting positive-probability existence.","tokens_in":10252,"feed_emoji":"⚖️","tokens_out":6760,"duration_ms":55687,"temperature":0.7,"pith_summary":"This paper proves a precise condition under which two sub-populations competing for a shared, accumulated resource can settle into a lasting equilibrium. If the ratio of immigrant to home population sizes converges to a finite positive limit, then both populations must survive indefinitely, and the resource threshold each generation uses converges to a unique level. That level is determined by a linear balance equation equating expected resource production with expected consumption in both groups. The proof also yields a constraint: at the threshold, the expected reproduction rates of the two sub-populations must be equal and at least one, and if that common rate is strictly larger than one, such an equilibrium can actually occur.","feed_headline":"Stable population ratio forces a unique resource threshold","feed_subtitle":"Proof shows two sub-populations coexist only when their reproduction rates match at the threshold.","key_machinery":"The central object is the resource-dependent branching process (RDBP) with two sub-populations drawing claims from the same accumulated resource space. The argument is carried by the limiting equilibrium equation, obtained by dividing the resource balance by the home population size, writing the immigrant contribution as $\\alpha$ times a per-capita term, and applying the strong law of large numbers conditioned on survival. The companion constraint $r_h F_h(F) = r_i F_i(F)$ emerges from taking limits on both sides of the recurrence expressing the next generation's ratio, and the Borel-Cantelli arguments control extinction and supercritical growth.","core_discovery":"The central claim is that if the ratio $\\Gamma^i_t/\\Gamma^h_t$ of immigrant to home population sizes converges almost surely to some $\\alpha \\in (0,\\infty)$, then both sub-processes tend to infinity almost surely and the resource threshold $F_t$ converges almost surely to a unique value $F := F(\\alpha)$. The limiting pair satisfies the equilibrium equation\n$$ r_h \\int_0^F x\\,dF_h(x) + \\$\\alpha$\\, r_i \\int_0^F x\\,dF_i(x) = R_h + \\$\\alpha$\\, R_i, $$\ntogether with the constraint qualification $r_h F_h(F) = r_i F_i(F) \\ge 1$. If the common value is strictly greater than one, then such an equilibrium exists with strictly positive probability. This is the content of Theorem 3 of the companion paper, for which the present supplement supplies the full proof.","pith_inferences":["The equilibrium equation suggests a direct calibration recipe: given an observed ratio $\\alpha$, the implied threshold $F(\\alpha)$ solves the balance equation, so the model yields a concrete, falsifiable prediction linking population ratios, resource production means, and claim distributions.","Because the proof requires all claims to be submitted to one shared resource space, the model would not apply to settings with segregated resource pools or priority access; testing those settings would need a modified equilibrium condition.","One could test the boundary case $r_h F_h(F) = r_i F_i(F) = 1$ numerically to see whether equilibria are indeed of probability zero, sharpening the paper's dichotomy.","The optional transport formulation suggests that policy interventions that alter claim distributions shift both $\\alpha$ and $F$; this could be used to compute the minimal-cost redistribution of claims needed to reach a target equilibrium."],"forward_implications":["If the paper is right, lasting coexistence of two sub-populations in this model requires both to be supercritical at the equilibrium threshold; a sub-population whose reproduction rate at the threshold is below one is driven to extinction.","The equilibrium threshold $F(\\alpha)$ is unique whenever the claim densities do not both vanish near it, so the eventual composition of the population is determined by the balance equation.","The condition $r_h F_h(F) = r_i F_i(F) \\ge 1$ is testable: it gives a concrete inequality that parameters and claim distributions must satisfy before any finite-positive-ratio equilibrium is possible.","With the strict inequality $>1$, the equilibrium is not just a limiting possibility but occurs with strictly positive probability, so the model predicts that such coexistence is a stable outcome.","The author indicates that the same approach, via the BRS-inequality, can be extended to an arbitrary number of sub-populations, though only the two-population case is fully proved here."],"supporting_citations":[{"why":"Supplies the resource-dependent branching process model and Theorem 4.4 ii) b) used to prove that supercritical sub-processes survive with positive probability.","marker":"Bruss and Duerinckx (2015)"},{"why":"Provides the Borel-Cantelli counterpart used in part (I) to show both sub-processes must tend to infinity if a finite positive ratio limit exists.","marker":"Bruss (1980)"},{"why":"The companion paper whose Theorem 3 is proved here; it states the equilibrium equation and constraint qualification.","marker":"Bruss (2024)"},{"why":"Supplies the extinction criterion used in the contradiction argument for necessity of the reproduction-rate constraint.","marker":"Bruss (1978)"},{"why":"Introduces the BRS-inequality used to indicate how the results extend to more than two sub-populations.","marker":"Bruss (2021)"}],"fun_headline_variants":["Ratio limit pins down unique resource threshold","Converging population ratio yields unique threshold","Unique resource threshold follows from stable ratio","Coexistence forces unique threshold in branching model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument relies on the assumption that all resource claims of both sub-populations are submitted to the same accumulated resource space, so total consumption is the additive sum of the two groups' claims; if the groups drew on separate pools or one had priority access, the equilibrium equation would be different.","fun_headline_variants_meta":{"raw":{"variants":["Ratio limit pins down unique resource threshold","Converging population ratio yields unique threshold","Unique resource threshold follows from stable ratio","Coexistence forces unique threshold in branching model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000525,"raw_usage":{"total_tokens":2485,"prompt_tokens":846,"completion_tokens":1639,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":1585}},"tokens_in":462,"tokens_out":1639,"duration_ms":11863,"temperature":1.0,"reasoning_tokens":1585,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T00:25:41.252822+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a two-population resource-dependent branching process satisfying all assumptions except that the two sub-populations draw from separate resource spaces; if a finite positive limiting ratio $\\alpha$ still exists, the claimed equilibrium equation need not hold. Alternatively, simulate the model with claim densities such that $r_h F_h(F) = r_i F_i(F) = 1$ and check whether the ratio never converges to a finite positive constant, contradicting positive-probability existence.","supporting_citations":[],"review_version":1}