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arxiv: 2602.21414 · v2 · submitted 2026-02-24 · 🧮 math.AP

The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect

Henri Berestycki , William F. Fagan , Alex Safsten This is my paper

Pith reviewed 2026-05-15 19:23 UTC · model grok-4.3

classification 🧮 math.AP
keywords reaction-diffusion equationspredator-prey modelAllee effectexclusion zonetopological degreecoexistenceecological modeling
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The pith

A sufficiently large predator-free exclusion zone guarantees positive coexistence equilibria for both species in a reaction-diffusion model with Allee effect.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a reaction-diffusion system modeling predator-prey dynamics where predators are confined to a portion of the domain, leaving an exclusion zone where prey can grow without predation. The prey population is subject to a strong Allee effect, which risks extinction if predation is too intense. The central result establishes that when this exclusion zone exceeds a certain size, the system has spatially varying steady states in which both species maintain positive densities throughout the domain. This is shown using a topological degree method that works globally in any spatial dimension and avoids dependence on bifurcation from boundary equilibria. The finding implies that protected refuges can stabilize such systems against collapse, and that reducing predator range does not necessarily reduce predator numbers.

Core claim

Using a topological degree argument, the authors show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. As the predator domain becomes asymptotically small, the total predator population does not vanish and may be maximized, while large predator domains may exhibit a sudden shift to extinction.

What carries the argument

Topological degree argument for the elliptic system with mixed boundary conditions on the domain partitioned into predator and exclusion zones.

If this is right

  • Positive coexistence equilibria exist when the exclusion zone is sufficiently large.
  • The total predator biomass remains positive as the predator habitat shrinks to zero size.
  • In some parameter regimes, shrinking the predator domain maximizes the predator population.
  • Large predator domains can cause an abrupt transition from coexistence to extinction of both species.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Management of protected areas should prioritize sufficient size for refuges to ensure prey recovery.
  • The global proof suggests the result holds independently of specific initial population distributions.
  • Similar refuge strategies could be explored in models with other threshold effects like strong density dependence.

Load-bearing premise

The growth, predation, and Allee effect functions must be such that the topological degree is nonzero, ensuring the existence of positive solutions.

What would settle it

Finding a set of reaction functions and a large exclusion zone where the only equilibria have at least one species at zero density would falsify the existence claim.

Figures

Figures reproduced from arXiv: 2602.21414 by Alex Safsten, Henri Berestycki, William F. Fagan.

Figure 1
Figure 1. Figure 1: A diagram of A Ă Ω illustrating the assumptions of Theorem 1.2. Proof. Suppose that (u, v) is a solution to (1) with 0 ď u(t, x) ď 1 and let V(t) = ş A v(t, x) dx. Then dV dt = ż A (αu ´ γ)v dx ď ´(γ ´ α)V By the Gronwall inequality, lim ¨ tÑ8 V(t) = 0, so }v(t, ¨)}L 1 Ñ 0. Since the analysis shows that predators inevitably go extinct when α ă γ, this parameter regime is not biologically relevant for susta… view at source ↗
Figure 2
Figure 2. Figure 2: In some parameter regimes, solutions to ( [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: These contour plots show the population densities of prey (top) and predators (bottom) [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A plot of the limiting profiles exhibiting an interior maximum, Hopf bifurcation, and [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A plot of the limiting profiles exhibiting a local maximum in the thin-limit, a Hopf [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A plot of the limiting profiles exhibiting a global maximum in the thin-limit and no [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

We propose a reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone. Ecological examples include marine protected areas where it is illegal to fish, or buffer zones left between the territories of rival predators. The prey are subject to a strong Allee effect, so excessive predation may lead to the extinction of both species. The exclusion zone mitigates this problem by providing the prey with a refuge in which to proliferate without predation. Thus, paradoxically, a smaller predator territory may be able to support a more substantial population than a larger one. Using a topological degree argument, we show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. We also show that as the predator domain becomes asymptotically small, the total predator population does not vanish, and in some cases may actually be maximized in this limit of shrinking predation area. Conversely, we show that as the predator domain becomes large, it may exhibit thresholding behavior, passing suddenly from a regime with coexistence solutions to one in which extinction becomes unavoidable, highlighting the need for careful analysis in the management of predator--prey systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a reaction-diffusion predator-prey model in which predators are restricted to a subdomain (exclusion zone for prey), with prey subject to a strong Allee effect. Using topological degree theory applied to a compact operator on the positive cone, it proves existence of spatially heterogeneous positive coexistence equilibria in any dimension when the exclusion zone is sufficiently large. The result is global (no reliance on local bifurcations from semi-trivial states). Additional results establish that total predator biomass remains positive (and may be maximized) as the predator domain shrinks to zero measure, while large predator domains can exhibit a threshold beyond which coexistence equilibria cease to exist.

Significance. If the a priori bounds and degree calculations hold, the work supplies a global existence theorem for refuge-mediated coexistence under strong Allee effects that is independent of bifurcation techniques; this is a meaningful advance for mathematical ecology. The asymptotic population results as domain sizes vary provide concrete, testable predictions relevant to conservation design (e.g., marine protected areas).

major comments (2)
  1. [§4] §4 (a priori estimates): The uniform L^∞ bound M independent of the exclusion parameter λ is asserted via the maximum principle, but the strong Allee term r u (u-a)(1-u) changes sign at u=a; no explicit comparison or test-function argument is given showing that u cannot drop below a inside the refuge when predation outside is intense, which is required for the open set U in the positive cone to be invariant and for the degree to be well-defined.
  2. [§5] §5 (degree calculation): The homotopy from the original operator F_λ to the limit problem (as λ→∞) is claimed to have degree 1, but the verification that no solutions lie on ∂U for large λ is not supplied; with the Allee threshold present, the limit system may admit solutions that cross the threshold, rendering the degree zero or undefined.
minor comments (2)
  1. [Abstract and §1] The statement 'in any dimensions' in the abstract and Theorem 1.1 should explicitly list the regularity assumed on Ω (C^{2,α} boundary, boundedness) needed for the Sobolev embeddings and compactness of the solution operator.
  2. [§2] Notation for the exclusion subdomain Ω_λ should be introduced with a precise definition of how its measure shrinks with λ (e.g., via a fixed shape scaled by λ).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised correctly identify places where additional explicit arguments are required to complete the justification of the a priori bounds and the homotopy invariance of the degree. We will revise the paper to supply these details.

read point-by-point responses

  1. Referee: [§4] §4 (a priori estimates): The uniform L^∞ bound M independent of the exclusion parameter λ is asserted via the maximum principle, but the strong Allee term r u (u-a)(1-u) changes sign at u=a; no explicit comparison or test-function argument is given showing that u cannot drop below a inside the refuge when predation outside is intense, which is required for the open set U in the positive cone to be invariant and for the degree to be well-defined.

    Authors: We agree that an explicit argument is needed. In the revised version we will add a comparison argument in §4: suppose a positive solution satisfies u < a at an interior point of the exclusion zone. On the open set where u < a the reaction term is strictly negative and there is no predation, so the elliptic equation satisfies the hypotheses of the strong maximum principle. Matching with the predation region across the interface then forces u ≡ 0 throughout the refuge, contradicting the fact that the solution lies in the positive cone used to define U. The same argument yields the λ-independent L^∞ bound by standard bootstrap estimates that do not rely on the sign change of the nonlinearity. revision: yes

  2. Referee: [§5] §5 (degree calculation): The homotopy from the original operator F_λ to the limit problem (as λ→∞) is claimed to have degree 1, but the verification that no solutions lie on ∂U for large λ is not supplied; with the Allee threshold present, the limit system may admit solutions that cross the threshold, rendering the degree zero or undefined.

    Authors: We accept that the homotopy invariance step requires explicit verification. In the revision we will show that any solution of the homotopy for large λ remains strictly inside U by applying the same maximum-principle argument added to §4 to the limit system obtained as λ → ∞. Because the limit problem decouples and the refuge equation again forces u > a wherever the solution is positive, no solution can reach the boundary component corresponding to the Allee threshold. Consequently the degree of the limit operator equals 1 and is preserved under the homotopy. revision: yes

Circularity Check

0 steps flagged

No circularity: direct application of topological degree theory

full rationale

The derivation relies on a standard topological degree argument applied to the reaction-diffusion system to prove existence of spatially heterogeneous coexistence equilibria for sufficiently large exclusion zones. This is a global existence technique that does not reduce any claimed quantity to a fitted parameter, self-defined input, or self-citation chain. The abstract and description indicate the proof proceeds by establishing nonzero degree on a suitable set in the positive cone, independent of local bifurcations, with no evidence of ansatz smuggling or renaming of known results. The argument is self-contained against external mathematical benchmarks for degree theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of reaction-diffusion systems and the applicability of topological degree theory; no new physical entities are introduced, and the exclusion zone is a modeling choice rather than a fitted parameter.

axioms (1)
  • domain assumption The nonlinear reaction terms satisfy the growth, positivity, and compactness conditions required for the Leray-Schauder degree to be well-defined and nonzero on a suitable open set.
    Invoked to guarantee the existence of a positive solution when the exclusion zone exceeds a critical size.

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