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arxiv: 2604.08132 · v1 · submitted 2026-04-09 · 🧮 math.DS

Dynamics of a Predator-Prey Model with Allee Effect and Interspecific Competition

Lina Peng , Jianhang Xie This is my paper

Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🧮 math.DS
keywords predator-prey modelAllee effectinterspecific competitionHolling response functionequilibrium stabilitybifurcation analysisLotka-Volterra systemJacobian eigenvalues
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The pith

A predator-prey model with Allee effect and interspecific competition yields boundary and internal equilibria that can be nodes, saddles, saddle-nodes or codimension-two cusp points.

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

The paper constructs two Lotka-Volterra predator-prey systems that incorporate an Allee effect and competition within the predator population, using Holling type II and type I response functions in turn. It determines the conditions on parameters such as growth rates, competition coefficients and carrying capacities under which different equilibrium points exist. These equilibria are then classified by the signs and values of the eigenvalues of the Jacobian matrix evaluated at each point. The resulting types include stable and unstable nodes, saddle points, saddle-node bifurcations and cusp points of codimension two. The work also isolates the precise parameter regimes in which an internal equilibrium has exactly one zero eigenvalue, a zero eigenvalue paired with purely imaginary eigenvalues, or a double zero eigenvalue.

Core claim

Boundary equilibrium points exist under a range of parameter conditions while internal equilibria appear only when certain additional inequalities on the parameters hold. At these points the equilibria may be stable or unstable nodes, saddle points, saddle-nodes, or cusp points with a codimension of 2. The paper supplies the explicit parameter conditions under which internal equilibria possess one zero eigenvalue and two non-zero eigenvalues, one zero eigenvalue together with a pair of purely imaginary eigenvalues, or two zero eigenvalues and one non-zero eigenvalue.

What carries the argument

The Jacobian matrix of the system evaluated at each equilibrium point, whose eigenvalues classify the local phase portrait and detect the listed degeneracies.

If this is right

  • For parameter values producing a stable internal node, both populations persist at positive constant densities.
  • Saddle equilibria imply that small perturbations can drive one population to extinction while the other survives.
  • Saddle-node bifurcations allow pairs of equilibria to appear or disappear abruptly when a parameter crosses a critical value.
  • Codimension-two cusp points organize more complex transitions, including the possibility of hysteresis in population levels.
  • Purely imaginary eigenvalues at an internal equilibrium indicate the potential for nearby periodic orbits.

Where Pith is reading between the lines

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

  • The two Holling forms produce overlapping but non-identical ranges of parameters that support stable coexistence, suggesting the functional response shape influences the size of the basin of attraction.
  • Adding stochastic noise or time delays to the same equations would likely turn the codimension-two cusps into more intricate bifurcation structures not captured here.
  • Field measurements of predator competition intensity could be mapped directly onto the derived inequalities to predict whether observed populations sit near a cusp or a simple saddle.

Load-bearing premise

The Allee effect and interspecific competition are represented by the particular functional forms chosen for the growth and interaction terms.

What would settle it

Numerical integration or field data for a predator-prey system with measured Allee threshold and competition strength that produces an internal equilibrium whose Jacobian eigenvalues fall outside the three listed zero-eigenvalue configurations.

Figures

Figures reproduced from arXiv: 2604.08132 by Jianhang Xie, Lina Peng.

Figure 1
Figure 1. Figure 1: Population density curves of three species when [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Population density curves of three species when condition (1) of Theorem 5 is satisfied [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
read the original abstract

This paper primarily discusses the dynamical properties of a class of Lotka-Volterra models featuring the Allee effect and interspecific competition within the predator population. The constructed models employ Holling II and Holling I response functions for the predator, respectively.The existence of boundary equilibrium points under various parameter conditions and internal equilibrium points under specific parameter conditions is discussed. The equilibrium points of the system may be stable or unstable nodes, saddle points, saddle-nodes, or cusp points with a codimension of 2. The parameter conditions under which internal equilibrium points possess one zero eigenvalue and two non-zero eigenvalues, one zero eigenvalue and a pair of purely imaginary eigenvalues, or two zero eigenvalues and one non-zero eigenvalue are analyzed.

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

3 major / 2 minor

Summary. The paper studies a 3D predator-prey ODE system that augments a Lotka-Volterra model with an Allee effect in the prey and interspecific competition within the predator, employing both Holling type I and type II functional responses. It establishes existence conditions for boundary equilibria and for positive internal equilibria, then classifies the local stability of these points as nodes, saddles, saddle-nodes, or codimension-2 cusps. Explicit parameter regimes are derived under which the Jacobian at an internal equilibrium has one zero eigenvalue (with the other two nonzero), one zero eigenvalue together with a purely imaginary pair, or a double zero eigenvalue (with the third nonzero).

Significance. If the claimed classifications of the degenerate equilibria are rigorously verified by center-manifold reductions and nondegeneracy conditions on the quadratic and cubic coefficients, the results would furnish concrete bifurcation diagrams and parameter thresholds for the onset of saddle-node and Bogdanov-Takens bifurcations in an ecologically motivated model, thereby contributing to the literature on Allee-effect-induced complexity in predator-prey dynamics.

major comments (3)
  1. [stability analysis of internal equilibria] The classification of internal equilibria possessing a double-zero eigenvalue as “cusp points with a codimension of 2” (abstract and the corresponding stability section) is based solely on the spectrum of the Jacobian. In a three-dimensional system this spectrum identifies the linear degeneracy but does not determine the normal-form coefficients; the quadratic and cubic terms on the two-dimensional center manifold must be computed and shown to satisfy the standard nondegeneracy conditions for a cusp (or Bogdanov-Takens) bifurcation. No such calculation appears to be supplied.
  2. [parameter conditions for eigenvalue patterns] The parameter conditions stated for the three listed eigenvalue patterns (one zero + two nonzero; one zero + imaginary pair; two zero + one nonzero) are obtained by solving the characteristic polynomial. However, the manuscript does not verify that the remaining transversality conditions (e.g., nonzero speed of the zero eigenvalue crossing the imaginary axis, or nondegeneracy of the unfolding parameters) hold in the claimed open sets of parameter space; without these, the local phase-portrait conclusions remain incomplete.
  3. [existence of internal equilibrium points] Existence of the internal equilibrium is asserted under “specific parameter conditions,” yet the algebraic conditions that guarantee a unique positive root of the resulting cubic (or quadratic) equilibrium equation are not displayed explicitly, nor is it shown that the Jacobian is well-defined and the equilibrium lies in the positive octant for those parameter values.
minor comments (2)
  1. [model formulation] Notation for the two distinct functional responses (Holling I versus Holling II) should be introduced with explicit equations rather than by reference to “respectively” in the abstract.
  2. [numerical simulations] Several parameter inequalities are stated without accompanying numerical examples or phase portraits that illustrate the claimed stability types; inclusion of at least one representative bifurcation diagram would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects needed to strengthen the rigor of the bifurcation analysis. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: The classification of internal equilibria possessing a double-zero eigenvalue as “cusp points with a codimension of 2” (abstract and the corresponding stability section) is based solely on the spectrum of the Jacobian. In a three-dimensional system this spectrum identifies the linear degeneracy but does not determine the normal-form coefficients; the quadratic and cubic terms on the two-dimensional center manifold must be computed and shown to satisfy the standard nondegeneracy conditions for a cusp (or Bogdanov-Takens) bifurcation. No such calculation appears to be supplied.

    Authors: We agree that the spectrum of the Jacobian alone is insufficient to classify the equilibrium as a codimension-2 cusp in a three-dimensional system. The manuscript identifies the double-zero eigenvalue but does not perform the center-manifold reduction or compute the quadratic and cubic coefficients to verify the nondegeneracy conditions required for the Bogdanov-Takens bifurcation. We will revise the manuscript to include this center-manifold analysis and the verification of the nondegeneracy conditions. revision: yes

  2. Referee: The parameter conditions stated for the three listed eigenvalue patterns (one zero + two nonzero; one zero + imaginary pair; two zero + one nonzero) are obtained by solving the characteristic polynomial. However, the manuscript does not verify that the remaining transversality conditions (e.g., nonzero speed of the zero eigenvalue crossing the imaginary axis, or nondegeneracy of the unfolding parameters) hold in the claimed open sets of parameter space; without these, the local phase-portrait conclusions remain incomplete.

    Authors: The stated parameter conditions were obtained by solving the characteristic equation for the desired eigenvalue configurations. We concur that transversality conditions, including nonzero crossing speeds and nondegeneracy of unfolding parameters, must be verified to ensure the bifurcations are generic in the claimed open parameter sets. In the revision we will explicitly check and document these transversality conditions for each of the three eigenvalue patterns. revision: yes

  3. Referee: Existence of the internal equilibrium is asserted under “specific parameter conditions,” yet the algebraic conditions that guarantee a unique positive root of the resulting cubic (or quadratic) equilibrium equation are not displayed explicitly, nor is it shown that the Jacobian is well-defined and the equilibrium lies in the positive octant for those parameter values.

    Authors: Existence of internal equilibria is claimed under specific parameter regimes obtained from the equilibrium equations. To make this rigorous, we will explicitly state the algebraic conditions that ensure a unique positive root of the cubic or quadratic, and we will verify that the resulting equilibrium lies in the positive octant with the Jacobian matrix well-defined at that point. These details will be added to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit ODE analysis with no fitted inputs or self-referential claims

full rationale

The manuscript writes down an explicit 3D ODE system (Lotka-Volterra with chosen Holling-type responses, Allee term, and competition term), solves algebraically for the coordinates of boundary and interior equilibria as functions of the parameters, computes the Jacobian at those points, and states the algebraic conditions on parameters that make the characteristic polynomial have one zero root, a zero-plus-imaginary pair, or a double-zero root. These steps are direct symbolic manipulations of the given vector field; they do not rename a fitted quantity as a prediction, invoke a self-citation for a uniqueness theorem, or smuggle an ansatz. The classification statements (nodes, saddles, saddle-nodes, codim-2 cusps) are asserted after the eigenvalue conditions, but the derivation itself reduces only to the model equations supplied at the outset. No load-bearing circular step exists.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the standard assumptions of autonomous ODE modeling in ecology; no new entities are introduced and no parameters are fitted to data.

axioms (1)
  • domain assumption The population dynamics are governed by a smooth autonomous system of ordinary differential equations with the stated functional forms for growth, predation, and competition.
    Invoked throughout the existence and stability analysis of equilibria.

pith-pipeline@v0.9.0 · 5413 in / 1291 out tokens · 43121 ms · 2026-05-10T17:47:34.839245+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

  1. [1]

    Sutherland

    Stephens, Philip A, William J. Sutherland. Consequences of the Allee effect for behavior, ecology, and conservation [J]. Trends in Ecology and Evolution, 1999, 14(10): 401-405

    work page 1999

  2. [2]

    Population dynamics with multiple Allee effects induced by fear factors –A mathematical study on prey-predator interactions[J]

    Sasmal S K. Population dynamics with multiple Allee effects induced by fear factors –A mathematical study on prey-predator interactions[J]. Applied Mathematical Modelling, 2018, 64: 1-14

    work page 2018

  3. [3]

    Dynamics of predator -prey model interaction with intraspecific competition[C]

    Ikbal M. Dynamics of predator -prey model interaction with intraspecific competition[C]. Journal of Physics: Conference Series. IOP Publishing, 2021, 1940(1): 012006

    work page 2021

  4. [4]

    Complex dynamics of a three -species predator–prey model with two nonlinearly competing species[J]

    Panja P, Gayen S, Kar T, et al. Complex dynamics of a three -species predator–prey model with two nonlinearly competing species[J]. Results in Control and Optimization, 2022, 8: 100153

    work page 2022

  5. [5]

    Dynamical Systems [M]

    Sternberg S. Dynamical Systems [M]. Courier Corporation, 2010

    work page 2010

  6. [6]

    Elements of Applied Bifurcation Theory [M]

    Kuznetsov Y A, Kuznetsov I A, Kuznetsov Y. Elements of Applied Bifurcation Theory [M]. New York: Springer, 1998

    work page 1998