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arxiv: 2508.09760 · v3 · submitted 2025-08-13 · 🧮 math.DS

Grazing duration and intensity modulate vegetation dynamics in semi-arid ecosystems with seasonal succession

Junhong Gan , Guohong Zhang , Xiaoli Wang This is my paper

Pith reviewed 2026-05-18 23:16 UTC · model grok-4.3

classification 🧮 math.DS
keywords vegetation dynamicssemi-arid ecosystemsgrazing managementpiecewise periodic modelseasonal successioncompetitive exclusionbistabilitydynamical systems
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The pith

A piecewise periodic model identifies duration thresholds for dry seasons and grazing that determine vegetation persistence, extinction, or competitive outcomes in semi-arid ecosystems.

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

The paper builds a mathematical model that splits each year into three phases—dry season, growth period, and grazing period in the wet season—to track how grazing affects plant populations in areas with strong seasonal shifts. It calculates specific lengths for the dry season and grazing period that mark the switch between a single plant species surviving year after year or dying out. When two plant species compete, the model shows that changing how long or how heavy the grazing is can produce one species taking over, both species living together steadily, or bistable states where either species can win depending on initial conditions. Simulations confirm these shifts by mapping out changes in behavior as grazing settings vary. The work aims to give practical guidance on timing and strength of grazing to keep vegetation stable in dry regions.

Core claim

In this piecewise periodic dynamical system, critical thresholds for the durations of the dry season and the grazing period separate persistence from extinction for a single vegetation species. For two competing species, the grazing parameters control the competitive outcomes of exclusion, coexistence, and bistability, with numerical simulations displaying the corresponding bifurcation diagrams and phase transitions.

What carries the argument

The novel piecewise periodic model that divides the annual cycle into three distinct phases with piecewise constant parameters in each phase.

If this is right

  • Dry season duration above a critical value leads to extinction of a single vegetation species while shorter durations allow persistence.
  • Grazing period length and intensity shift competitive outcomes among exclusion, coexistence, and bistability for two species.
  • Bifurcation diagrams from the model display phase transitions as grazing regimes change.
  • The thresholds supply concrete targets for adjusting grazing timing to support vegetation in arid and semi-arid regions.

Where Pith is reading between the lines

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

  • If the thresholds hold in the field, managers could extend or shorten grazing periods in specific years to steer outcomes toward coexistence rather than loss of species.
  • Adding rainfall variability or stochastic events to the phase structure could test whether the same thresholds remain reliable under climate fluctuations.
  • The phase-division method might transfer to other seasonal systems such as crop rotations or wildlife population cycles with distinct activity periods.

Load-bearing premise

The ecosystem can be represented accurately by splitting the year into three phases with parameters that stay constant inside each phase and reflect real ecological processes.

What would settle it

Long-term field measurements in a semi-arid site showing that vegetation persists when the dry season length exceeds the model's critical threshold or goes extinct when it falls short of that threshold.

Figures

Figures reproduced from arXiv: 2508.09760 by Guohong Zhang, Junhong Gan, Xiaoli Wang.

Figure 1
Figure 1. Figure 1: Seasonal succession with three distinct time period. The duration of the bad season is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The evolution of u(t) and v(t) when one of conditions of Theorem 3.4 are satisfied. Example 4.2. (a) Given the parameters d1 = 0.1, d2 = 0.5, b1 = 0.2, b2 = 0.2, r = 1, c1 = 0.4, c2 = 0.6, τ1 = 6, τ2 = 8, T = 10, the thresholds are τ ∗ 2 = 1.5 and τ ∗∗ 2 = 8.33, which implies that τ1 < τ ∗ 1 and τ ∗ 2 < τ2 < τ ∗∗ 2 . It follows from Theorem 3.2 that the unique semi-trivial T-periodic solution (u ∗ (t), 0) … view at source ↗
Figure 3
Figure 3. Figure 3: Stability of two semitrival pereidic solution ( [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Coexistence of two species u(t) and v(t). There exists a globally asymptotically stable positive periodic solution (¯u(t), v¯(t)) originating from (¯u, v¯). 0 10 20 30 40 50 60 70 80 90 100 Time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The population density u(t) v(t) 0 10 20 30 40 50 60 70 80 90 100 Time 0 0.2 0.4 0.6 0.8 1 The population density u(t) v(t) [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bistability of two semitrival periodic solution ( [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effects of the duration of dry season and grazing season on the dynamics of the single [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effects of the duration of dry season and grazing season on the dynamics of the competition [PITH_FULL_IMAGE:figures/full_fig_p037_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effects of the duration of dry season and grazing season on the dynamics of the competition [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
read the original abstract

This study investigates the impacts of grazing duration and intensity on vegetation population dynamics in semi-arid ecosystems characterized by seasonal succession. A novel piecewise periodic model is proposed, dividing the annual cycle into three distinct phases: dry season, growth period and grazing period in wet season. We derive critical thresholds for the durations of the dry season and grazing period that determine the persistence or extinction of a single vegetation species. For two competing species, we analyze how grazing parameters influence competitive outcomes, including exclusion, coexistence, and bistability. Theoretical results are supported by numerical simulations, which illustrate bifurcation diagrams and phase transitions under varying grazing regimes. Our findings provide actionable insights for sustainable grazing management in arid and semi-arid regions.

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 proposes a piecewise periodic model dividing the annual cycle into three phases (dry season, growth period, grazing period) with constant parameters in each. It derives explicit critical thresholds for dry-season and grazing-period durations governing single-species persistence or extinction via the period map. For two competing species it analyzes grazing-parameter effects on exclusion, coexistence, and bistability, with results illustrated by numerical bifurcation diagrams and phase-transition plots.

Significance. If the thresholds are correctly obtained from the integrated growth factors of the period map, the work supplies a transparent mathematical link between grazing timing/intensity and vegetation outcomes that could inform management in semi-arid systems. The explicit derivation and bifurcation analysis constitute a clear strength; however, the overall significance hinges on whether the piecewise-constant idealization produces thresholds that remain informative under more gradual environmental transitions.

major comments (2)
  1. [§2] §2 (Model formulation): the central thresholds are obtained from the product of exponential growth factors across the three fixed-length intervals with instantaneous switches. Because real moisture, temperature, and grazing intensity vary continuously rather than abruptly, the effective integrals differ and the location of the critical-duration curves in parameter space can shift; this assumption is load-bearing for the persistence/extinction claims.
  2. [§4] §4 (Numerical results): the bifurcation diagrams and phase portraits are generated entirely inside the same piecewise-constant framework. They therefore cannot detect the discrepancy that would arise from smoothed transitions, leaving the robustness of the reported threshold curves untested.
minor comments (2)
  1. [§3] Notation for the period map and the integrated growth factors could be introduced earlier and used consistently to improve readability of the threshold derivations.
  2. [§4] The manuscript would benefit from a brief sensitivity check (e.g., replacing one abrupt switch with a short linear ramp) to quantify how much the critical durations move.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which highlight important considerations regarding the model's assumptions. We agree that the piecewise-constant idealization is a simplification of real-world gradual transitions, and we will revise the manuscript to address the robustness of our findings. We respond to each major comment below.

read point-by-point responses

  1. Referee: §2 (Model formulation): the central thresholds are obtained from the product of exponential growth factors across the three fixed-length intervals with instantaneous switches. Because real moisture, temperature, and grazing intensity vary continuously rather than abruptly, the effective integrals differ and the location of the critical-duration curves in parameter space can shift; this assumption is load-bearing for the persistence/extinction claims.

    Authors: We recognize that the abrupt switches in our piecewise periodic model represent an idealization, as real environmental variables change gradually. This simplification enables the explicit computation of the period map and the derivation of critical thresholds for dry-season and grazing-period durations. To strengthen the manuscript, we will add a new subsection discussing the potential impact of smoothed transitions on the threshold locations and include a brief numerical sensitivity analysis comparing the piecewise model to one with continuous transition functions. revision: partial

  2. Referee: §4 (Numerical results): the bifurcation diagrams and phase portraits are generated entirely inside the same piecewise-constant framework. They therefore cannot detect the discrepancy that would arise from smoothed transitions, leaving the robustness of the reported threshold curves untested.

    Authors: The numerical results, including bifurcation diagrams, are indeed computed within the piecewise-constant model to illustrate the theoretical findings. We agree that this does not test robustness to gradual changes. In the revision, we will incorporate additional simulations using smoothed transition functions for the seasonal phases and compare the resulting bifurcation structures and critical curves to those in the original model. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from piecewise model equations

full rationale

The paper constructs a piecewise periodic dynamical system dividing the year into three phases with constant rates, then derives persistence/extinction thresholds and competitive outcomes explicitly from the period map and Floquet multipliers of that system. These steps follow standard analytic techniques for periodic ODEs and do not reduce to fitted data, self-citations, or tautological redefinitions. Numerical bifurcation diagrams are produced inside the same model and serve only as illustration, not as independent validation that would create circularity. No load-bearing self-citation or ansatz smuggling is present in the provided derivation chain.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the construction of the piecewise model and the analysis of its equilibria and stability, which involve several rate parameters that are not derived from first principles but chosen to represent the system. The main axioms are the division into three phases and the form of the population growth and grazing terms.

free parameters (2)
  • growth rates in different phases
    Likely chosen or calibrated to represent ecological conditions in each season for the thresholds to be meaningful.
  • grazing intensity and duration parameters
    Central parameters varied to explore management scenarios and determine competitive outcomes.
axioms (2)
  • domain assumption The ecosystem dynamics can be modeled using ordinary differential equations with piecewise periodic coefficients.
    Standard approach in mathematical ecology for systems with seasonal variation.
  • ad hoc to paper Grazing occurs only during a defined period in the wet season with uniform intensity.
    Core modeling choice enabling the three-phase piecewise structure.

pith-pipeline@v0.9.0 · 5647 in / 1531 out tokens · 52455 ms · 2026-05-18T23:16:50.795781+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Feng, Y.F

    X.M. Feng, Y.F. Liu, S.G. Ruan, J.S. Yu, Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting. J. Differ. Equ. (2023) , 354:237-263

    work page 2023

  2. [2]

    Hsu and X.Q

    S.B. Hsu and X.Q. Zhao, A Lotka-Volterra competition model with seasonal suc- cession. J. Math. Biol. (2012) , 64(1-2):109-130

    work page 2012

  3. [3]

    Peng, X.Q

    R. Peng, X.Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. (2013), 33: 2007-2031

    work page 2013

  4. [4]

    L. Pu, Z. Lin, Y. Lou, A West Nile virus nonlocal model with free boundaries and seasonal succession, J. Math. Biol. (2023), 86: 1-52

    work page 2023

  5. [5]

    L. Niu, Y. Wang, X. Xie, Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications, Discrete Contin. Dyn. Syst., Ser. B (2021), 26: 2161-2172

    work page 2021

  6. [6]

    Sommer, Z.M

    U. Sommer, Z.M. Gliwicz, W. Lampert, A. Duncan, The PEG-model of seasonal succession of planktonic events in fresh waters, Arch. Hydrobiol. (1986), 106: 433- 471

    work page 1986

  7. [7]

    M. Wang, Q. Zhang, X.Q. Zhao, Dynamics for a diffusive competition model with seasonal succession and different free boundaries, J. Differ. Equ. (2021), 285: 536- 582

    work page 2021

  8. [8]

    Peng, X.Q

    R. Peng, X.Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity(2012), 25 :1451-1471

    work page 2012

  9. [9]

    Wang, X.Q

    X. Wang, X.Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math.(2017), 77: 181-201

    work page 2017

  10. [10]

    S. Liu, Y. Lou, Classifying the level set of principal eigenvalue for time-periodic parabolic operators and applications, J. Funct. Anal. 282 (2022) 109338

    work page 2022

  11. [11]

    Browne, X

    C.J. Browne, X. Pan, H. Shu, X. Wang, Resonance of periodic combination an- tiviral therapy and intracellular delays in virus model, Bull. Math. Biol. (2020), 82: 1-29. 39

    work page 2020

  12. [12]

    Liu, X.M

    Y.F. Liu, X.M. Feng, S.G. Ruan, and J.S. Yu. Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting, II: Existence of two periodic solutions. J. Differ. Equ. (2024) , 388:253-285

    work page 2024

  13. [13]

    Liu, J.S

    Y.F. Liu, J.S. Yu, and J. Li. Global dynamics of a competitive system with seasonal succession and different harvesting strategies. J. Differ. Equ. (2024) , 382:211-245

    work page 2024

  14. [14]

    J. Chen, J. Huang, S. Ruan, J. Wang, Bifurcations of invariant tori in predator- prey models with seasonal prey harvesting, SIAM J. Appl. Math. (2013), 73: 1876- 1905

    work page 2013

  15. [15]

    Darabsah, Y

    I.A. Darabsah, Y. Yuan, A stage-structured mathematical model for fish stock with harvesting, SIAM J. Appl. Math. (2018), 78: 145-170

    work page 2018

  16. [16]

    W. Gan, Z. Lin, M. Pedersen, Delay-driven spatial patterns in a predator-prey model with constant prey harvesting, Z. Angew. Math. Phys. (2022), 73: 1-18

    work page 2022

  17. [17]

    Xiao, Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discrete Contin

    D. Xiao, Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst. Ser. B (2016), 21: 699-719

    work page 2016

  18. [18]

    Xiao, L.S

    D. Xiao, L.S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math. (2005), 65: 737-753

    work page 2005

  19. [19]

    Yu and J

    J.S. Yu and J. Li, Global asymptotic stability in an interactive wild and sterile mosquito model. J. Differ. Equ. (2020) , 269(7):6193-6215

    work page 2020

  20. [20]

    Yu, Existence and stability of a unique and exact two periodic orbits for an interactive wild and sterile mosquito model

    J.S. Yu, Existence and stability of a unique and exact two periodic orbits for an interactive wild and sterile mosquito model. J. Differ. Equ. (2020) , 269: 10395- 10415

    work page 2020

  21. [21]

    Zheng, J

    B. Zheng, J. Li, J. Yu, One discrete dynamical model on Wolbachia infection frequency in mosquito populations. Sci. China Math. (2022) , 65: 1749-1764

    work page 2022

  22. [22]

    Zheng, J

    B. Zheng, J. Yu, J. Li, Modeling and analysis of the implementation of the Wol- bachia incompatible and sterile insect technique for mosquito population suppression. SIAM J. Appl. Math. (2021) , 81: 718-740

    work page 2021

  23. [23]

    Eigentler, J.A

    L. Eigentler, J.A. Sherratt, Effects of precipitation intermittency on vegetation patterns in semi-arid landscape, Physica D. (2020), 405: 132396. 40

    work page 2020

  24. [24]

    Eigentler, ¡ ¤ J.A

    L. Eigentler, ¡ ¤ J.A. Sherratt, An integrodifference model for vegetation patterns in semi-arid environments with seasonality, J. Math. Biol. (2020), 81 (3):875-904

    work page 2020

  25. [25]

    Wang, G.H

    X.L. Wang, G.H. Zhang, Vegetation pattern formation in seminal systems due to internal competition reaction between plants, J. Theor. Biol. (2018), 458:10-14

    work page 2018

  26. [26]

    Xiaoli Wang, Guohong Zhang, The influence of infiltration feedback on the char- acteristic of banded vegetation pattern on hillsides of semiarid area, Plos One( 2019),14(1): e0205715

    work page 2019

  27. [27]

    J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: Patterns with the largest possible propagation speeds, Proc. R. Soc. A(2011), 467: 3272-3294

    work page 2011

  28. [28]

    C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Sci- ence(1999), 284:1826-1828

    work page 1999

  29. [29]

    Rietkerk, S. C. Dekker, P. C. de Ruiter, and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems, Science(2004), 305: 1926-1929

    work page 2004

  30. [30]

    M. K. Pal and S. Pori, Effect of nonlocal grazing on dry-land vegetation dynamics, Phys. Review. E. (2022), 106: 054407

    work page 2022

  31. [31]

    Maimaiti, W.B

    Y. Maimaiti, W.B. Yang, Spatial vegetation pattern formation and transition of an extended water ¨Cplant model with nonlocal or local grazing, Nonlinear Dyn. (2024), 112 5765-5791

    work page 2024

  32. [32]

    Siero, Nonlocal grazing in patterned ecosystem, J

    E. Siero, Nonlocal grazing in patterned ecosystem, J. Theor. Biol. (2018), 436: 64-71

    work page 2018

  33. [33]

    Pan, H.Y

    X.J. Pan, H.Y. Shu, L. Wang, X.S. Wang, J.S. Yu On the periodic solutions of switching scalar dynamical systems J. Differ. Equ. (2025) , 415: 365-382. 41

    work page 2025