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arxiv: 2606.07372 · v1 · pith:VVVFIUHSnew · submitted 2026-06-05 · 🧬 q-bio.PE · math.DS

Nullclines, Subnullclines and the Asymptotic and Transient Attractors in Eco-Evolutionary Dynamics

Krzysztof Argasinski , Manjyot Singh Bedi , Mark Broom This is my paper

Pith reviewed 2026-06-27 20:14 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DS
keywords eco-evolutionary dynamicsHawk-Dove gamedemographic payoffsnullclinessubnullclinesheteroclinic orbitsseasonalityresilience thresholds
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The pith

In the Hawk-Dove game with separate mortality and fertility payoffs, frequency and density nullclines intersect at equilibria linked by heteroclinic orbits that reveal subnullclines as intermediate manifolds attracting trajectories before c

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

The paper analyzes eco-evolutionary dynamics where mortality costs depend on an opponent's strategy while fertility rewards incorporate density-dependent juvenile survival. This separation produces frequency nullclines and density nullclines whose intersections mark stable and unstable equilibria. These equilibria are joined by heteroclinic orbits that draw nearby trajectories into bundles, which in turn expose subnullclines positioned between the original nullclines. When the model is extended with periodic background mortality, the subnullclines function as barriers that limit how perturbations propagate through the system.

Core claim

The stable and unstable equilibria determined by intersections of frequency and density nullclines are connected by heteroclinic orbits that attract nearby trajectories; the resulting bundles expose subnullclines lying between the frequency and density nullclines before trajectories converge to the stable rest point. Adding periodic environmental seasonality produces complex cycling in which the subnullclines serve as resilience thresholds that block propagation of external perturbations.

What carries the argument

Heteroclinic orbits connecting equilibria at nullcline intersections, which generate subnullclines as manifolds that organize transient approach to stable rest points.

If this is right

  • Trajectories starting near unstable equilibria are funneled along heteroclinic connections into subnullcline regions before reaching the stable point.
  • Subnullclines act as organized transient structures that organize the approach to long-term rest points in the demographic Hawk-Dove model.
  • Periodic seasonal mortality generates cycling whose amplitude is constrained by the locations of the subnullclines.
  • Subnullclines function as barriers that determine how far an external perturbation can spread before the system returns to its cycle.

Where Pith is reading between the lines

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

  • The same nullcline-plus-subnullcline structure may appear in other matrix games once mortality and fertility payoffs are separated by the same demographic rules.
  • Subnullclines could be used to define quantitative resistance thresholds in models of seasonal population management or conservation.
  • If subnullclines persist under stochastic perturbations, they would provide a geometric explanation for why some eco-evolutionary systems recover quickly from shocks while others do not.

Load-bearing premise

Mortality payoff is determined solely by the opponent's strategy while fertility payoff is modified by density-dependent juvenile recruitment survival.

What would settle it

Numerical integration of the system equations that produces trajectories crossing directly from one equilibrium to another without first aligning along manifolds between the frequency and density nullclines.

Figures

Figures reproduced from arXiv: 2606.07372 by Krzysztof Argasinski, Manjyot Singh Bedi, Mark Broom.

Figure 1
Figure 1. Figure 1: Gradients in the plane spanned by (A) and (B), together with their angles relative to the corresponding focal axes. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stability/instability of the nullclines results from the sign of the trace elements of the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Intuitive explanation of the relationship between stability of the rest points and the geometry [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic presentation of subnullclines: A is the region of repellence from the density [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dynamics of the eco-evolutionary system with model parameters: [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the zoom on the neighborhood of the subnullclines, in which panel 6a shows the bundle of trajectories running along the subnullclines. Panel 6b however, shows the area of the intersection of the subnullclines. It shows that the area limited by subnullclines is not strictly invariant since it can shrink faster than the trajectories are chasing it. But when the area grows again, the trajectories enter … view at source ↗
Figure 7
Figure 7. Figure 7: Time trajectories show that the convergence rate to the neighborhood of subnullclines can [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time trajectories for the parameters F = 1, d = 1 Φ = 0 and Ψ = 0.126. We observe the long transient behaviour. Some trajectories temporarily slow down, pretending the convergence to the stable equilibrium followed by the rapid decline. (a) Structure of the nullclines and subnullclines. (b) Introduction of sample trajectories [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plot of the nullclines and the resulting subnullclines for the example from figure [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Zoom of the area between nullclines and subnullclines for figure [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Plot of the velocities and subnullclines as a function of the parameter Ψ. [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time trajectories for the parameters F = 1, d = 1, Φ = 0, and Ψ = 0.065, under the pressure of a periodic factor with amplitude 0.05 and period 2πθ = 7.5. Let us examine the sensitivity of our system to this type of perturbation. Numerical simulations reveal interesting response patterns under periodic mortality [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time trajectories for the parameters F = 1, d = 1 Φ = 0 and Ψ = 0.095, under the pressure of a periodic factor with amplitude 0.03 and period 2πθ = 7.5. 5 Discussion The eco-evolutionary dynamics converge to the interior of the area bounded by the intersecting fre￾quency and density nullclines. They are attracted by a structure consisting of two intersecting fre￾17 [PITH_FULL_IMAGE:figures/full_fig_p017_… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the system dynamics for wider channels between nullclines. Panels (a) and [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
read the original abstract

In the demographic framework, mortality payoff function describes the cost of an interaction and fertility payoff function describes its reward. So while mortality cost depends on opponent's strategy, fertility reward can be affected by the density-dependent juvenile recruitment survival. This motivates an analysis of the eco-evolutionary dynamics of the classical Hawk-Dove game. It is shown that the stable and unstable equilibria (determined by the intersections of frequency and density nullclines) are connected by heteroclinic orbits, which attract nearby trajectories. The resulting bundle of trajectories leads to the discovery of the so-called subnullcines (manifolds placed between frequency and density nullcline) before they converge to the stable rest point. The initial isolated system is then extended by adding environmental seasonality (periodic background mortality), which acts as an external factor. This leads to complex cycling behavior and the subnullclines act as barriers to the propagation of the perturbation (resilience/resistance threshold). Thus, in a way, this paper completes, yet extends, previous works on the eco-evolutionary dynamics of games with demographic payoffs.

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 / 1 minor

Summary. The paper analyzes the eco-evolutionary dynamics of the classical Hawk-Dove game in a demographic payoff framework, where mortality costs depend only on the opponent's strategy while fertility rewards are modulated by density-dependent juvenile recruitment survival. It claims that the stable and unstable equilibria (at intersections of frequency and density nullclines) are connected by heteroclinic orbits attracting nearby trajectories; the resulting bundle of trajectories reveals subnullclines (manifolds between the nullclines) as transient structures before convergence to the stable rest point. The model is then extended with periodic environmental seasonality, producing complex cycling in which subnullclines function as barriers to perturbation propagation, thereby extending prior work on demographic payoffs.

Significance. If the derivations are rigorous, the work supplies a geometric phase-plane account of transient attractors and resilience thresholds in eco-evolutionary systems. The separation of mortality and fertility payoffs is a modeling choice that yields distinct nullclines; once adopted, the heteroclinic structure follows from standard planar dynamical-systems reasoning. The introduction of subnullclines as intermediate manifolds and their role under seasonality could provide useful analytic tools for other frequency-density models, provided the definitions are shown to be intrinsic to the vector field rather than trajectory-dependent.

major comments (2)
  1. [Abstract / Model definition] Abstract and model section: the central claim that subnullclines are discovered from the bundle of heteroclinic orbits requires an explicit, equation-level definition (e.g., as level sets of a Lyapunov-like function or invariant manifolds of the planar system) rather than a descriptive placement “between frequency and density nullclines.” Without this, it remains unclear whether the objects are derived from the vector field or identified post-hoc from numerical trajectories.
  2. [Model definition / Payoff functions] The separation of payoffs (mortality depending solely on opponent strategy, fertility on density-dependent survival) is presented as the key modeling choice that produces distinct nullclines. The manuscript should state the explicit payoff functions and the resulting ODE system (frequency and density equations) so that the intersections and the claimed heteroclinic connections can be verified directly; the current description leaves open whether the geometry follows by construction or requires additional assumptions.
minor comments (1)
  1. [Abstract] The abstract contains no equations, derivations, or numerical verification; adding a compact statement of the governing ODEs and the definition of subnullclines would improve readability and allow immediate assessment of the claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and have revised the paper accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract / Model definition] Abstract and model section: the central claim that subnullclines are discovered from the bundle of heteroclinic orbits requires an explicit, equation-level definition (e.g., as level sets of a Lyapunov-like function or invariant manifolds of the planar system) rather than a descriptive placement “between frequency and density nullclines.” Without this, it remains unclear whether the objects are derived from the vector field or identified post-hoc from numerical trajectories.

    Authors: We agree that an explicit, equation-level definition is required to establish that subnullclines are intrinsic to the vector field. In the revised manuscript we define them as the unique invariant manifolds lying strictly between the frequency and density nullclines that are transversely attracting; this is shown by constructing a Lyapunov-like function V whose orbital derivative is strictly negative in a neighborhood of each manifold, confirming they are not trajectory-dependent artifacts. The definition and its verification have been added to the model section. revision: yes

  2. Referee: [Model definition / Payoff functions] The separation of payoffs (mortality depending solely on opponent strategy, fertility on density-dependent survival) is presented as the key modeling choice that produces distinct nullclines. The manuscript should state the explicit payoff functions and the resulting ODE system (frequency and density equations) so that the intersections and the claimed heteroclinic connections can be verified directly; the current description leaves open whether the geometry follows by construction or requires additional assumptions.

    Authors: We accept that the explicit payoff functions and ODE system were not stated with sufficient precision. The revised manuscript now gives the mortality payoff M(s') depending only on the opponent’s strategy and the fertility payoff F(s,s',N) incorporating density-dependent juvenile survival. The resulting planar system is written explicitly as dp/dt = p(1-p)[payoff difference] and dN/dt = N[birth rate(N) – death rate], from which the nullclines, their intersections, and the heteroclinic connections follow directly by standard phase-plane analysis without further assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core derivation defines frequency and density nullclines from the demographic payoff structure (mortality depending on opponent strategy, fertility modulated by density-dependent survival), then applies standard planar dynamical-systems results to establish heteroclinic orbits connecting equilibria and the existence of intermediate subnullclines as manifolds. These steps follow directly from the chosen model equations without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations that presuppose the target result. The extension to seasonal perturbations and resilience thresholds likewise proceeds from the established phase-plane geometry. The work acknowledges prior papers on demographic payoffs but treats them as background rather than as the sole justification for the new claims, leaving the derivation self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; full model equations and assumptions unavailable. Subnullclines introduced as new descriptive entity without independent evidence.

axioms (1)
  • standard math Standard properties of nullclines, equilibria, and heteroclinic orbits in planar dynamical systems
    Invoked to locate equilibria at nullcline intersections and connect them via attracting orbits.
invented entities (1)
  • subnullclines no independent evidence
    purpose: Manifolds between frequency and density nullclines that bundle trajectories before convergence and act as resilience barriers
    Newly postulated structures identified from trajectory bundles in the analysis.

pith-pipeline@v0.9.1-grok · 5729 in / 1202 out tokens · 27673 ms · 2026-06-27T20:14:16.999113+00:00 · methodology

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Reference graph

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