{"id":"67ab6420-e70c-4000-bf77-a9459785a9b4","arxiv_id":"2505.03727","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For organisms that diffuse and randomly reset to a fixed location, the critical patch half-width is ℓc = arccos(r cos(xr/√(1−r))) / √(1−r) in a lethal exterior, and resetting can either raise or lower the required habitat size.","lead":"A new analytical formula describes how the minimum habitat size needed for survival changes when organisms intermittently return to a fixed spot. The formula shows that this resetting can shrink or grow the required habitat depending on where and how often the return happens.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (18) for partial relocation with reset outside the patch fails the r→0 limit: as printed it does not reproduce the reset-free critical size tan ℓc = √a, so the dashed curves in Figs. 7(b,c) and parts of Figs. 8–9 are unreliable.","rationale":"The paper's abstract and central qualitative claim — that stochastic resetting can increase or decrease the critical habitat size depending on reset rate, reset position, and exterior hostility — are supported for the totally hostile case and for xr inside the patch, where Eq. (7) is self-consistent and simulations in Figs. 2 and 4 show good agreement. The reader's identified concern about Eq. (7) being used when |xr| < ℓc fails is not as decisive as stated: the paper does introduce Eq. (9) for the outside-patch branch, notes that this branch requires |xr| > π/2, and explicitly excludes reduction for |xr| > ℓc* in the phase diagram. The limits ℓc(1,xr) = √(2+xr²) and ℓc(∞,xr) = |xr| are also derivable for parameter regions where xr lies inside the patch, so the reader's asymptotic objection does not land cleanly. However, Eq. (18) contains a clear algebraic error that invalidates a substantial part of the quantitative analysis: it does not reduce to the known r = 0 result, which is an unambiguous internal test. This affects the partial-relocation results for reset positions outside the habitat, including dashed curves in Figs. 7(b,c) and the corresponding phase-diagram boundaries in Figs. 8 and 9. Because these figures are presented as quantitative predictions and the equation as printed cannot reproduce the reset-free limit, the paper in its current form is not reliable. The error appears to be a missing factor of r in one term, suggesting a bounded fix, but the required correction touches multiple figures and numerical results, so the appropriate outcome is rejection pending major revision rather than acceptance as is.","tokens_in":14842,"tokens_out":36996,"duration_ms":333966,"concrete_test":"Set r = 0, a = 1, xr = 2 in Eq. (18) and substitute the reset-free critical value ℓc = arctan(√a) = π/4 ≈ 0.785; the equation is not satisfied (substitution gives ≈ −1.56, not zero), whereas tan ℓc = √a should hold identically. Then independently re-derive the determinant condition for xr ∈ Ωout with the delta jump at xr and verify whether it yields a(γ cos ℓc − sin ℓc) = r sin ℓc e^{−γ(xr−ℓc)}; if so, regenerate the dashed curves in Fig. 7(b) with this corrected equation, or run agent-based simulations at a = 1, r = 0.5, xr = 2, and compare the simulated critical half-width with the two predictions.","verdict_should_be":"REJECT","load_bearing_attack":"The most load-bearing concern is not the Eq. (7) domain issue emphasized by the reader, which the paper partially self-answers with Eq. (9) and by restricting the reduction region to |xr| < π/2. The stronger, internally checkable problem is Eq. (18), used in Section IV B for partial relocation when xr ∈ Ωout. At r = 0, resetting is absent, so the critical half-width must reduce to the standard reset-free condition tan ℓc = √a, independent of xr. Setting r = 0 in Eq. (18) gives e^{γxr} a(γ cos ℓc − sin ℓc) = e^{γℓc} sin ℓc, which is not satisfied by tan ℓc = √a (e.g., a = 1, xr = 2, ℓc = π/4 gives a nonzero left-hand side and a root near 0.64 instead of 0.785). Re-deriving the stationary determinant in Appendix C with the delta-jump condition at xr yields a(γ cos ℓc − sin ℓc) = r sin ℓc e^{−γ(xr−ℓc)}, i.e., the printed Eq. (18) is missing the factor r in the second term. Consequently the dashed curves for xr outside the patch in Figs. 7(b,c), the outside-patch portions of the phase diagram in Fig. 8, and the corresponding panels of Fig. 9 are based on an incorrect equation and cannot be trusted. The reader's objection to Eq. (7) is real but secondary; the r→1 and r→∞ limits quoted for xr inside are derivable, whereas Eq. (18) has a direct internal inconsistency.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the critical patch size for a one-dimensional population that grows inside a finite habitat, dies in the exterior, and is subject to stochastic resetting to a fixed position x_r. The authors derive analytical expressions for the critical half-width \\ell_c for two exterior scenarios (totally hostile and finitely hostile) and for two reset protocols (total relocation from anywhere, and partial relocation from the exterior only). They report three qualitative regimes in the (x_r, r) plane: resetting reduces the critical size for any rate, reduces it only above a threshold rate, or never reduces it. Agent-based simulations are used to validate the transient dynamics, stationary profiles, and in some cases the critical sizes. The paper claims that resetting can either increase or decrease the required habitat size depending on the reset position, the reset rate, and the external mortality rate.","tokens_in":15187,"tokens_out":34512,"duration_ms":292729,"significance":"If correct, the paper provides a useful and parameter-free extension of the classical critical-patch-size problem to a class of movement behavior (stochastic resetting) that is relevant for homing, refuge-seeking, and central-place foraging. The analytical derivations are explicit, and the agent-based simulations serve as a consistency check of the same stochastic rules used in the derivation. However, two load-bearing issues prevent the results from being relied upon in their current form: the outside-patch partial-relocation equation does not reduce to the reset-free limit, and the displayed Eq. (7) is inconsistent with its own derivation and with the limits quoted immediately after it. Because these errors affect parts of the phase diagrams and the corresponding figures, the manuscript requires substantial correction before its central quantitative claims can be accepted.","major_comments":[{"comment":"Eq. (18) fails the r -> 0 limit, which is the central consistency test for the partial-relocation model. Setting r = 0 in Eq. (18) with gamma = sqrt(a) gives e^{sqrt(a) x_r} a (sqrt(a) cos ell_c - sin ell_c) = e^{sqrt(a) ell_c} sin ell_c. This is not solved by the reset-free critical condition tan ell_c = sqrt(a); for a = 1, x_r = 2, ell_c = pi/4 the left side vanishes while the right side does not. The derivation in Appendix C must therefore contain an algebraic error in the determinant for x_r in Omega_out; the term e^{gamma ell_c} sin ell_c should carry an additional factor of r (or, equivalently, the displayed equation is missing a reset-rate factor). Consequently the dashed curves for x_r outside the patch in Figs. 7(b,c), the outside-patch boundaries in Fig. 8, and the corresponding lower-row panels of Fig. 9 are based on an incorrect equation and need to be recomputed from the correctly derived transcendental condition.","section":"Section IV B, Eq. (18)"},{"comment":"The formula displayed as Eq. (7), with x_r / sqrt(1-r) inside the cosine, is inconsistent with the derivation in Appendix B. The stationary solution in Appendix B yields cos(ell_c sqrt(1-r)) = r cos(x_r sqrt(1-r)), so the argument of the cosine should be x_r sqrt(1-r), not x_r / sqrt(1-r). With the printed form, the claimed resonant limit ell_c(1, x_r) = sqrt(2 + x_r^2) does not follow, because the argument of the cosine oscillates as r -> 1, and the claimed asymptotic value ell_c(infinity, x_r) = |x_r| is not obtained either (the printed expression tends to 0 for fixed x_r). The authors should correct Eq. (7) and re-examine every subsequent limit and figure that relies on it, including the curves in Fig. 3(a).","section":"Section III A, Eq. (7)"},{"comment":"Even after correcting the algebraic form of Eq. (7), the solution is only valid when the reset position lies inside the patch, |x_r| < ell_c(r, x_r). The paper acknowledges this condition and introduces Eq. (9) for the outside-patch case, but Fig. 3(a) plots solid curves from Eq. (7) over parameter ranges where the resulting ell_c is smaller than |x_r| (for example, the |x_r| = 1.9 and 2.5 curves at small r). The text and figures should make explicit whether Eq. (7), Eq. (9), or no survival (for r >= 1 with x_r outside) applies in each part of the parameter plane; as printed, the solid branches in those ranges are not physical critical sizes.","section":"Section III A, domain of validity"}],"minor_comments":[{"comment":"The delta-source term r N_Omega_out delta(x - x_r) is displayed in both the interior and exterior equations, but it should appear only in the equation for the region that contains x_r. This typo should be corrected because it affects the readability of the subsequent derivation.","section":"Section IV B, Eqs. (16)"},{"comment":"The single dashed curve labeled Eq. (9) is easily misread as applying globally; actually the switch from Eq. (7) to Eq. (9) occurs at an r value that depends on x_r. Marking the crossover point on each solid curve would remove the ambiguity.","section":"Fig. 3(a)"},{"comment":"The statement that in the limit a -> infinity ell_c becomes independent of r is correct for Eq. (17) with x_r inside the patch, but it should be stated with that restriction, since the dashed outside-patch branches in the same figure are the ones affected by the Eq. (18) error.","section":"Section IV B, paragraph after Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The core idea is timely and the inside-patch results are likely correct, but the paper as submitted has more than one load-bearing inconsistency: Eq. (18) fails a basic limit, and the displayed Eq. (7) does not match its own derivation. These are localized algebraic errors that can in principle be fixed by re-deriving the stationary determinants and recomputing the affected figures, so I recommend major revision rather than rejection. The authors should also add simulation support for the outside-patch curves in Figs. 7-9, which currently contain no symbols and are exactly the curves affected by the Eq. (18) error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a major-revision paper, not a reject. The core idea—stochastic resetting in the critical patch-size problem—is new, and the closed-form expressions for reset positions inside the patch are likely correct. The backward-master-equation approach is standard but competently used, and the agent-based simulations match the theory in the regimes where the theory is valid. The qualitative message that resetting can either shrink or enlarge the required habitat, depending on rate, reset location, and exterior hostility, is plausible and well supported inside the valid parameter region.\n\nThe load-bearing problem is Eq. (18), for partial relocation when the reset position is outside the patch. The cleanest check is the r→0 limit: with no resetting, the critical size must reduce to the standard tan ℓc = √a. The printed equation does not; setting r=0 leaves an extra term that only vanishes when sin ℓc=0. Re-deriving the stationary determinant from the jump condition at xr gives a(γ cos ℓc − sin ℓc) = r sin ℓc e^{−γ(xr−ℓc)}, i.e., the second term in Eq. (18) is missing a factor of r. This means the dashed curves in Figs. 7(b,c), the outside-patch parts of the Fig. 8 phase diagram, and the corresponding panels of Fig. 9 are based on an incorrect equation and cannot be trusted. This is not a cosmetic typo; it changes the predicted regimes.\n\nThe second issue is the domain of validity of Eq. (7). It assumes the reset position lies inside the patch, but the paper plots it for |xr| values where ℓc drops below |xr|. The text notes the switch to Eq. (9) but does not enforce it in Fig. 3(a), so the curves for |xr|=1.9 and 2.5 contain unphysical small-r branches. The claimed limit ℓc(∞,xr)=|xr| also deserves scrutiny; it may be a branch artifact of the complex arccosine rather than a real limiting behavior.\n\nThe good news is that both problems are fixable without changing the overall framework. The r→0 check is trivial and should have been done; once Eq. (18) is corrected and the phase diagrams redrawn with the validity region marked, the core results for xr inside the patch and the general method stand up. I'd send this to peer review, but the referee should insist on the corrected equation and a clear statement of where Eq. (7) applies.\n\nWho this is for: researchers in stochastic resetting and spatial ecology, and anyone who wants a worked example of why you always check a transcendental equation against the zero-coupling limit.","headline":"Worth a major revision, not a desk reject: the closed-form results for resetting inside the patch are real, but a missing factor of r in Eq. (18) invalidates the partial-relocation phase diagrams, and the validity domain of Eq. (7) is not tracked.","tokens_in":15742,"tokens_out":6655,"would_cite":false,"duration_ms":61942,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Stochastic resetting can shrink or enlarge the minimum habitat a population needs to survive, and the paper gives a closed formula that says which.","keywords":["stochastic resetting","critical patch size","population persistence","reaction-diffusion equation","habitat fragmentation","extinction threshold","diffusion with resetting","agent-based simulation"],"falsifier":"Take parameters where the predicted critical patch cannot contain the reset site, for instance $x_r=1.0$ and $r$ large enough that Eq. (7) gives $\\ell_c<1$; simulate the agent-based process there and compare survival to the outside-reset prediction. Agreement with Eq. (7) would validate its unstated domain of validity; either outcome settles whether the phase diagrams hold beyond $|x_r|<\\ell_c$.","tokens_in":14577,"feed_emoji":"🌿","tokens_out":10140,"duration_ms":89959,"temperature":0.7,"pith_summary":"This paper asks how much habitat a diffusing population needs when each organism occasionally jumps back to a fixed reset point, modeling homing or refuge-seeking. It derives the critical half-width $\\ell_c$ of a one-dimensional patch as a function of the reset rate $r$, the reset position $x_r$, and the hostility of the surrounding environment. The central result is an explicit formula for $\\ell_c$ in the absorbing-boundary case and closed or transcendental conditions for finite outside mortality, under two reset protocols. The answer is not monotone: resetting can lower the required habitat size, raise it, or leave it unchanged depending on where the reset point sits. This matters for conservation because movement behavior, not just patch area, sets the extinction threshold.","feed_headline":"Returning home can shrink or expand a species' minimum habitat","feed_subtitle":"A new formula maps when returning to a home site helps a population persist and when it backfires.","key_machinery":"The engine of the derivation is the backward master equation for the total population, $\\partial_t N(x_0)=\\partial_{x_0}^2 N(x_0)+(1-r)N(x_0)+rN(x_r)$, where $x_0$ is the initial position. Setting the dominant long-time growth rate $s_M$ to zero turns this into a stationary boundary-value problem whose self-consistency at $x_r$ produces Eq. (7); the same pole condition appears through Laplace inversion of the full time-dependent solution. For finite outside hostility, the equivalent machinery is a piecewise exponential and sinusoidal stationary solution together with continuity, flux, and delta-jump conditions, whose determinant condition yields the closed formulas and transcendental equations.","core_discovery":"For a perfectly hostile exterior, the paper's main claim is that the critical patch half-width is $\\ell_c(r,x_r)=\\arccos(r\\cos(x_r/\\sqrt{1-r}))/\\sqrt{1-r}$, recovering $\\pi/2$ when $r=0$ and $\\sqrt{2+x_r^2}$ at $r=1$. The slope at $r=0$ is $\\pi/4-\\cos x_r$, which yields three regimes: resetting helps at any rate when $|x_r|<\\arccos(\\pi/4)\\approx0.667$; it helps only above a threshold rate when $0.667<|x_r|<\\pi/2$; and it never helps when the reset site lies beyond $\\pi/2$. The paper extends this to finite outside mortality $a$, giving closed-form critical sizes when resets return organisms to the patch and transcendental equations when the reset site lies outside, for both total and outside-only relocation. Agent-based simulations reproduce the predicted population growth, stationary profiles, and critical sizes.","pith_inferences":["Editor's inference: the three-regime picture should carry over to two-dimensional patches, with the reset point replaced by a central refuge and the threshold condition set by the patch radius; the same eigenvalue argument applies to the radial Laplacian.","Editor's inference: under logistic growth, the linearized critical condition near extinction is unchanged, so the formulas should still mark the extinction boundary even with carrying capacity, a testable prediction.","Editor's inference: a distribution of reset positions would smooth the sharp regime boundaries but should preserve the central qualitative result that near-center refuges shrink habitat requirements while far refuges enlarge them."],"forward_implications":["When organisms reset to the patch center, the critical habitat size decreases monotonically with reset rate, so frequent homing can let a population persist in fragments far smaller than the reset-free threshold.","For reset sites between $x_r^*\\approx0.667$ and $\\pi/2$, weak resetting is harmful and only rates above a threshold $r^*$ reduce the required habitat, so a little homing can be worse than none.","With a perfectly hostile exterior and a reset site outside the patch, resetting cannot reduce the critical size; it only drains population unless $r<1$, in which case survival needs a larger patch.","In weakly hostile environments, resetting to a point just outside the patch can lower the critical size at a finite optimal rate, so the value of a refuge depends on the mortality contrast across the boundary.","The critical size does not depend on the initial spatial distribution, because the evolution equation is linear and arbitrary initial conditions are superpositions of point sources."],"supporting_citations":[{"why":"Supplies the original diffusion-with-stochastic-resetting framework and the reaction-diffusion equation with the delta source used in Eq. (3).","marker":"[21]"},{"why":"Provides the backward master equation and Laplace-inversion approach used to derive Eq. (5).","marker":"[25]"},{"why":"Gives the classic critical-size baseline $\\ell_c=\\pi/2$ that the paper recovers at $r=0$.","marker":"[10]"},{"why":"Provides the known $r=0$ total-population solution against which the no-reset limit is checked.","marker":"[28]"},{"why":"Motivates the analogy between resetting and restoring forces, where increasing the reset rate plays a role similar to increasing stiffness in movement-bias models.","marker":"[18]"}],"fun_headline_variants":["Stochastic resetting resizes species' survival zones","Homing behavior alters critical habitat size","Reset rate decides if returning home helps or hurts","Critical habitat size shifts with stochastic resetting","When returning home helps or hurts survival"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the reset location $x_r$ lies inside the viable patch, so every reset returns an organism to the habitat; if the formula predicts a patch too small to contain $x_r$, the reset source term disappears and Eq. (7) no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic resetting resizes species' survival zones","Homing behavior alters critical habitat size","Reset rate decides if returning home helps or hurts","Critical habitat size shifts with stochastic resetting","When returning home helps or hurts survival"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2340,"prompt_tokens":899,"completion_tokens":1441,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1374}},"tokens_in":515,"tokens_out":1441,"duration_ms":8920,"temperature":1.0,"reasoning_tokens":1374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:45:40.589263+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take parameters where the predicted critical patch cannot contain the reset site, for instance $x_r=1.0$ and $r$ large enough that Eq. (7) gives $\\ell_c<1$; simulate the agent-based process there and compare survival to the outside-reset prediction. Agreement with Eq. (7) would validate its unstated domain of validity; either outcome settles whether the phase diagrams hold beyond $|x_r|<\\ell_c$.","supporting_citations":[{"cited_title":"Neicu, A","cited_arxiv_id":null,"evidence_quote":"Supplies the original diffusion-with-stochastic-resetting framework and the reaction-diffusion equation with the delta source used in Eq. (3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the backward master equation and Laplace-inversion approach used to derive Eq. (5)."},{"cited_title":"Cantrell and C","cited_arxiv_id":null,"evidence_quote":"Gives the classic critical-size baseline $\\ell_c=\\pi/2$ that the paper recovers at $r=0$."},{"cited_title":"Ballard, V","cited_arxiv_id":null,"evidence_quote":"Motivates the analogy between resetting and restoring forces, where increasing the reset rate plays a role similar to increasing stiffness in movement-bias models."}],"review_version":1}