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arxiv: 2606.26340 · v1 · pith:BFODBAIInew · submitted 2026-06-24 · 🧬 q-bio.PE

An Investigation of Additional Food Models with Generalised Functional Response

Urvashi Verma , Kanishka Goyal , Saptarshi Biswas , James I. Lathrop , Rana D. Parshad This is my paper

Pith reviewed 2026-06-26 00:39 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords additional foodpredator-preyglobal stabilityBogdanov-Takens bifurcationchemical reaction network theoryHolling type IVdeficiencypest control
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The pith

Additional food models with generalized responses have globally stable coexistence equilibria under stated conditions and exhibit a codimension-3 Bogdanov-Takens bifurcation for Holling type IV.

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

The paper studies a general class of predator-prey systems that incorporate an extra food source for the predator alongside a generalized functional response. It derives conditions on the response functions that guarantee the equilibrium at which both populations persist is globally asymptotically stable. The analysis then specializes to a Holling type IV response and locates a Bogdanov-Takens bifurcation point of codimension three. Chemical reaction network theory is applied to the same models, showing that the addition of the food term raises the deficiency of the network and thereby may account for the observed complex bifurcations.

Core claim

For a general class of additional food models, conditions are established under which the coexistence equilibrium is globally stable. Focusing on Holling type IV functional response with additional food, the existence of a Bogdanov-Takens bifurcation of codimension 3 is shown. The introduction of additional food increases the deficiency of the underlying reaction network, suggesting a link between higher deficiency and complex bifurcations.

What carries the argument

Generalized functional response that includes an additional food term, with global stability proved via Lyapunov functions and deficiency computed via chemical reaction network theory.

If this is right

  • When the stated conditions on the response hold, every positive initial condition converges to the coexistence equilibrium.
  • The Holling type IV additional-food system admits a Bogdanov-Takens bifurcation of codimension exactly 3 at certain parameter values.
  • Adding the food source strictly increases the deficiency index of the underlying reaction network.
  • Higher deficiency is offered as the structural reason for the appearance of the codimension-3 bifurcation.

Where Pith is reading between the lines

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

  • If the deficiency-bifurcation link is general, then other structural modifications that raise deficiency should likewise produce higher-codimension bifurcations.
  • Pest-control strategies that add food might be tuned to stay inside the globally stable regime and thereby avoid unwanted oscillations.
  • Numerical continuation software could be used to track the codimension-3 point as the amount of additional food is varied.

Load-bearing premise

The functional responses and additional food terms must satisfy the positivity, monotonicity and boundedness properties needed for the Lyapunov arguments and the standard deficiency formulas to apply directly.

What would settle it

A concrete parameter set in the Holling type IV additional-food model for which either the coexistence equilibrium fails to be globally stable or no codimension-3 Bogdanov-Takens point exists would falsify the respective claims.

Figures

Figures reproduced from arXiv: 2606.26340 by James I. Lathrop, Kanishka Goyal, Rana D. Parshad, Saptarshi Biswas, Urvashi Verma.

Figure 1
Figure 1. Figure 1: If the numerical response g(x, ξ, α) is increasing on 0 < x < γ, then a unique coexistence equilibrium exist. If the form of the numerical response g(x, ξ, α) is non-monotonic on 0 < x < γ, then more than one coexistence equilibrium may exist. Proof. At a coexistence equilibrium (x ∗ , y∗ ), we have x ∗ > 0, y ∗ > 0, and dy dt = y(βg(x, ξ, α) − δ) = 0 Thus, y ∗ (βg(x ∗ , ξ, α) − δ) = 0 Since y ∗ > 0, it fo… view at source ↗
read the original abstract

Additional food sources are often used to improve the effectiveness of predators in controlling pest populations. However, the non-symmetric structure of additional food predator-prey models can cause certain aspects of their dynamics challenging to analyze. In this work, we study a general class of additional food models and establish conditions under which the coexistence equilibrium is globally stable. We then focus on a Holling type IV functional response with AF and show the existence of a Bogdanov-Takens bifurcation of codimension 3. We also study these models through the lens of deterministic chemical reaction network theory. Our analysis shows that the introduction of additional food increases the deficiency of the underlying reaction network and suggests a possible link between higher deficiency and complex bifurcations.

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 examines a general class of additional food (AF) predator-prey models with generalised functional responses. It derives conditions under which the coexistence equilibrium is globally stable. For the specific Holling type IV response with AF, it establishes the existence of a codimension-3 Bogdanov-Takens bifurcation. The models are also analysed via deterministic chemical reaction network theory (CRNT), where the introduction of AF is shown to increase network deficiency, with a suggested link between higher deficiency and the appearance of complex bifurcations.

Significance. If the global-stability conditions and the codim-3 BT result are rigorously derived, the work supplies concrete analytic tools for a class of models used in biological control. The CRNT perspective offers a potentially novel explanatory angle, but only if the deficiency calculations are shown to be well-defined for the non-mass-action kinetics employed.

major comments (3)
  1. [CRNT analysis section (and abstract)] The central CRNT claim (that AF increases deficiency and thereby promotes complex bifurcations) is load-bearing for the interpretive part of the paper. Generalised functional responses such as Holling type IV are not mass-action; the manuscript must specify exactly how the effective reaction network is constructed (auxiliary species, effective rates, etc.) and confirm that the standard deficiency theorems still apply. Without this, the suggested link between deficiency and the observed BT bifurcation lacks rigorous grounding.
  2. [Global stability section] The abstract asserts the existence of global-stability conditions for the general class, yet supplies no derivation outline, Lyapunov function, or statement of the precise technical hypotheses (positivity, monotonicity, boundedness) required for the proof. These hypotheses are listed as the weakest assumption in the reader’s report; the manuscript must state them explicitly and verify that the chosen generalised responses satisfy them.
  3. [Bifurcation analysis section] The codimension-3 Bogdanov-Takens bifurcation for the Holling type IV + AF model is a strong claim. The paper must exhibit the normal-form coefficients, the transversality conditions, and the parameter values at which the degeneracy occurs; merely stating existence without these calculations leaves the result unverifiable from the given information.
minor comments (2)
  1. [Notation and model formulation] Notation for the generalised functional response should be introduced once and used consistently; any re-definition in later sections should be cross-referenced.
  2. [Figures] Figure captions for phase portraits or bifurcation diagrams should explicitly list the parameter values used so that the codim-3 point can be reproduced.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We agree that additional details are needed to make the CRNT construction, global-stability hypotheses, and bifurcation calculations fully rigorous and verifiable. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [CRNT analysis section (and abstract)] The central CRNT claim (that AF increases deficiency and thereby promotes complex bifurcations) is load-bearing for the interpretive part of the paper. Generalised functional responses such as Holling type IV are not mass-action; the manuscript must specify exactly how the effective reaction network is constructed (auxiliary species, effective rates, etc.) and confirm that the standard deficiency theorems still apply. Without this, the suggested link between deficiency and the observed BT bifurcation lacks rigorous grounding.

    Authors: We accept that the network construction for non-mass-action kinetics must be stated explicitly. In the revision we will add a dedicated subsection describing the auxiliary species, the effective rate functions used to embed the generalised responses, and the verification that the deficiency is well-defined and that the standard CRNT deficiency theorems remain applicable. We will also clarify that the link to the BT bifurcation is suggestive rather than a direct theorem. revision: yes

  2. Referee: [Global stability section] The abstract asserts the existence of global-stability conditions for the general class, yet supplies no derivation outline, Lyapunov function, or statement of the precise technical hypotheses (positivity, monotonicity, boundedness) required for the proof. These hypotheses are listed as the weakest assumption in the reader’s report; the manuscript must state them explicitly and verify that the chosen generalised responses satisfy them.

    Authors: The referee correctly notes that the abstract and main text lack an explicit outline of the proof. We will insert a new subsection that states the precise technical hypotheses (positivity, monotonicity, boundedness), presents the Lyapunov function, and verifies that all generalised responses considered in the paper satisfy the hypotheses. A brief derivation sketch will also be added. revision: yes

  3. Referee: [Bifurcation analysis section] The codimension-3 Bogdanov-Takens bifurcation for the Holling type IV + AF model is a strong claim. The paper must exhibit the normal-form coefficients, the transversality conditions, and the parameter values at which the degeneracy occurs; merely stating existence without these calculations leaves the result unverifiable from the given information.

    Authors: We agree that the codimension-3 claim requires the explicit normal-form coefficients, transversality conditions, and the concrete parameter values at which the degeneracy is attained. In the revised version we will include these calculations (currently performed but not displayed) together with the numerical parameter set, rendering the result verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper establishes global stability conditions for the coexistence equilibrium via Lyapunov or similar methods and demonstrates a codimension-3 Bogdanov-Takens bifurcation through standard bifurcation analysis. These steps rely on technical assumptions about the generalized functional responses (positivity, monotonicity, boundedness) that are stated as inputs rather than derived from the outputs. The CRNT deficiency observation is presented only as a separate suggestion of a possible link, not as a load-bearing premise that derives the stability or bifurcation results. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citation chains appear in the claims. The central results remain independent of the CRNT interpretation and are grounded in external mathematical techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit list of free parameters, background axioms, or new entities; all such elements remain unidentified.

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Reference graph

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