pith. sign in

arxiv: 2605.18489 · v1 · pith:DWLV2MVTnew · submitted 2026-05-18 · 🧮 math.DS

Addendum to "Persistence and extinction in an Elk-Wolf prey-predator system with refuge and inter-regional movement. Appl. Math. Comput. 514 (2026) 129834"

Rajesh Das , Dibakar Ghosh , Sourav Kumar Sasmal This is my paper

Pith reviewed 2026-05-20 08:09 UTC · model grok-4.3

classification 🧮 math.DS
keywords elk-wolf modelprey-predator systemrefugeHopf bifurcationLyapunov functionstability analysisboundednessaddendum
0
0 comments X

The pith

Re-evaluation supplies corrected Lyapunov function and transversality criteria for the elk-wolf refuge model.

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

This addendum re-examines the elk-wolf prey-predator model with refuge and inter-regional movement that was put forward in an earlier paper. It identifies mistakes in the original local and global stability proofs, particularly the construction of the Lyapunov function, and supplies fixes that restore mathematical consistency. The note also clarifies boundedness of solutions, model formulation details, and the existence of equilibria. It rederives the conditions for Hopf bifurcation using the proper transversality criteria and checks the sensitivity and numerical results for reproducibility. The corrections are offered so that later studies of predator-prey systems that incorporate refuges can rest on reliable analytic statements.

Core claim

The remark re-evaluates the local and global stability analyses of the elk-wolf model, especially the construction of the Lyapunov function, and rederives the Hopf bifurcation conditions using the proper transversality criteria. It supplies clarifications on boundedness, model formulation, and the existence of equilibria while confirming that the sensitivity and numerical results remain consistent after the corrections.

What carries the argument

The corrected Lyapunov function used to establish global stability together with the rederived transversality condition that locates Hopf bifurcations in the elk-wolf system.

If this is right

  • The system now possesses mathematically consistent conditions for both local and global stability.
  • Hopf bifurcations occur only when the revised transversality criterion is satisfied.
  • Boundedness of solutions is established on firmer ground, supporting reliable statements about persistence and extinction.
  • Sensitivity analysis and numerical simulations become reproducible under the corrected formulation.
  • Future work on refuge-based predator-prey models can cite these consistent analytic statements without repeating the earlier errors.

Where Pith is reading between the lines

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

  • The corrections may reduce the chance that conservation models based on the elk-wolf system reach incorrect conclusions about population thresholds.
  • Similar re-examinations of Lyapunov constructions and transversality conditions could be performed on other published prey-predator models that use refuges or movement between patches.
  • The emphasis on proper transversality criteria draws attention to a frequent source of error when ecological models are analyzed for oscillatory behavior.

Load-bearing premise

The original analyses contained errors in the Lyapunov function construction and the application of transversality criteria that, once corrected, change the stability and bifurcation conclusions.

What would settle it

A direct calculation showing that the time derivative of the original Lyapunov function is not negative semi-definite along trajectories, or a verification that the original transversality condition fails to hold at the critical parameter value.

Figures

Figures reproduced from arXiv: 2605.18489 by Dibakar Ghosh, Rajesh Das, Sourav Kumar Sasmal.

Figure 1
Figure 1. Figure 1: Figure shows the existence and transversality conditions for occurance of the Hopf bifurcation [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Figure shows the sensitivity of system ( [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Figure interprets the solution trajectories of system ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Figure shows the stability region of the coexistence equilibrium [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Figure shows the bifurcation diagram of system ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Figure shows the occurrence of a stable limit cycle around the coexistence equilibrium [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
read the original abstract

The elk - wolf model with movements between refuge and open habitat was put forward in Maji et al. (Appl. Math. Comput. 514 (2026)), which is rigorously re-examined in this remark. We re-evaluate the local and global stability analyses, especially the construction of the Lyapunov function, and provide mathematical clarifications on boundedness, model formulation, and the existence of equilibria. The sensitivity and numerical results are re-examined for consistency and re-producibility, and the Hopf bifurcation conditions are rederived using the proper transversality criteria. The purpose of this note is to support future studies of predator-prey systems based on refuges by offering mathematically consistent conditions.

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

1 major / 1 minor

Summary. This addendum re-examines the elk-wolf prey-predator model with refuge and inter-regional movement originally presented in Maji et al. (Appl. Math. Comput. 514, 2026). It corrects errors in the Lyapunov function construction for global stability, clarifies boundedness proofs and model formulation details, re-derives the existence of equilibria, re-evaluates sensitivity and numerical results for consistency and reproducibility, and rederives Hopf bifurcation conditions using proper transversality criteria to yield mathematically consistent stability and bifurcation results.

Significance. If the corrections hold, the addendum supplies a reliable mathematical foundation for the refuge-movement elk-wolf system, enabling future studies of predator-prey dynamics with spatial refuges to proceed from accurate stability and bifurcation conditions. The explicit re-examination of numerical results and sensitivity analysis for reproducibility is a notable strength that supports verifiable applied mathematical ecology.

major comments (1)
  1. [Abstract and Hopf bifurcation re-derivation section] The central corrections to the Lyapunov function and transversality condition for Hopf bifurcation are load-bearing for the claim of mathematically consistent conditions, yet the manuscript presents these as direct replacements without displaying the full corrected derivative calculations or the explicit verification that the new transversality quantity changes sign at the critical parameter value (see abstract paragraph on re-evaluation of analyses and the section on Hopf bifurcation).
minor comments (1)
  1. [Numerical results section] The re-examined numerical simulations should list the exact parameter sets and initial conditions used in each figure to facilitate direct reproducibility checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review, as well as the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and Hopf bifurcation re-derivation section] The central corrections to the Lyapunov function and transversality condition for Hopf bifurcation are load-bearing for the claim of mathematically consistent conditions, yet the manuscript presents these as direct replacements without displaying the full corrected derivative calculations or the explicit verification that the new transversality quantity changes sign at the critical parameter value (see abstract paragraph on re-evaluation of analyses and the section on Hopf bifurcation).

    Authors: We agree that the full derivative calculations for the corrected Lyapunov function and the explicit verification that the transversality quantity changes sign at the critical value are necessary for rigor. In the revised manuscript we will expand the Hopf bifurcation section to include these complete step-by-step calculations and the sign-change verification, while retaining the corrected conditions already stated in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The addendum re-derives local and global stability results, constructs a corrected Lyapunov function, re-proves boundedness, clarifies model formulation and equilibria, and rederives Hopf bifurcation conditions using proper transversality criteria. These steps are presented as independent mathematical corrections to the prior Maji et al. work, with explicit replacements for flawed arguments rather than any reduction of outputs to inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citations. The central claims rest on direct re-evaluation of the differential equations and standard stability theorems, remaining self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard results from dynamical systems theory for stability and bifurcation analysis; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard theorems on local and global stability via Lyapunov functions for autonomous ODE systems.
    Invoked when re-evaluating the stability analyses and Lyapunov function construction.
  • standard math Transversality condition for Hopf bifurcation in planar or higher-dimensional systems.
    Used to rederive the Hopf bifurcation conditions.

pith-pipeline@v0.9.0 · 5669 in / 1293 out tokens · 68452 ms · 2026-05-20T08:09:46.640615+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

120 extracted references · 120 canonical work pages

  1. [1]

    The evolution of predator-prey interactions: theory and evidence , author=. Annu. R. Ecol. Syst. , volume=. 2000 , publisher=

    work page 2000

  2. [2]

    Chaos Solitons Frac

    Coefficient criterion for four-dimensional Hopf bifurcations: a complete mathematical characterization and applications to economic dynamics , author=. Chaos Solitons Frac. , volume=. 2003 , publisher=

    work page 2003

  3. [3]

    Hydra effect and global dynamics of predation with strong

    Bai, Dingyong and Wu, Jianhong and Zheng, Bo and Yu, Jianshe , journal=. Hydra effect and global dynamics of predation with strong. 2024 , publisher=

    work page 2024

  4. [4]

    Journal of Theoretical Biology , volume=

    Role of infection on the stability of a predator--prey system with several response functions—a comparative study , author=. Journal of Theoretical Biology , volume=. 2007 , publisher=

    work page 2007

  5. [5]

    British journal of cancer , volume=

    Investigating prostate cancer tumour--stroma interactions: clinical and biological insights from an evolutionary game , author=. British journal of cancer , volume=. 2012 , publisher=

    work page 2012

  6. [6]

    Epidemiology & Infection , volume=

    A clarification of transmission terms in host-microparasite models: numbers, densities and areas , author=. Epidemiology & Infection , volume=. 2002 , publisher=

    work page 2002

  7. [7]

    Ecology , volume=

    The orgins and evolution of predator-prey theory , author=. Ecology , volume=. 1992 , publisher=

    work page 1992

  8. [8]

    CA: a Cancer Journal for Clinicians , volume=

    Global cancer statistics 2022: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries , author=. CA: a Cancer Journal for Clinicians , volume=. 2024 , publisher=

    work page 2022

  9. [9]

    Trends in Ecology & Evolution , volume=

    Experimental coevolution of species interactions , author=. Trends in Ecology & Evolution , volume=. 2013 , publisher=

    work page 2013

  10. [10]

    International Journal of Bifurcation and Chaos , volume=

    Bifurcations and pattern formation in a predator--prey model , author=. International Journal of Bifurcation and Chaos , volume=. 2018 , publisher=

    work page 2018

  11. [11]

    Nonlinear Analysis , volume=

    A predator-prey model with disease in the prey , author=. Nonlinear Analysis , volume=. 1999 , publisher=

    work page 1999

  12. [12]

    Ecological Modelling , volume=

    Viral infection on phytoplankton--zooplankton system—a mathematical model , author=. Ecological Modelling , volume=. 2002 , publisher=

    work page 2002

  13. [13]

    Ecological Modelling , volume=

    Pelicans at risk in Salton Sea—an eco-epidemiological model-II , author=. Ecological Modelling , volume=. 2003 , publisher=

    work page 2003

  14. [14]

    Ecology Letters , volume=

    The interaction between predation and competition: a review and synthesis , author=. Ecology Letters , volume=. 2002 , publisher=

    work page 2002

  15. [15]

    Nature , volume=

    The interaction between predation and competition , author=. Nature , volume=. 2008 , publisher=

    work page 2008

  16. [16]

    Journal of theoretical biology , volume=

    Eco-evolutionary cyclic dominance among predators, prey, and parasites , author=. Journal of theoretical biology , volume=. 2023 , publisher=

    work page 2023

  17. [17]

    Journal of Ornithology , volume=

    Predation in bird populations , author=. Journal of Ornithology , volume=. 2011 , publisher=

    work page 2011

  18. [18]

    Trends in Ecology & Evolution , volume=

    Relationships between direct predation and risk effects , author=. Trends in Ecology & Evolution , volume=. 2008 , publisher=

    work page 2008

  19. [19]

    Coevolution can reverse predator--prey cycles , author=. Proc. Nat. A. Sci. , volume=. 2014 , publisher=

    work page 2014

  20. [20]

    Trends in Ecology & Evolution , volume=

    Parasites, disease and the structure of ecological communities , author=. Trends in Ecology & Evolution , volume=. 1986 , publisher=

    work page 1986

  21. [21]

    1934 , publisher=

    The struggle for existence , author=. 1934 , publisher=

    work page 1934

  22. [22]

    The American Naturalist , volume=

    Do hares eat lynx? , author=. The American Naturalist , volume=. 1973 , publisher=

    work page 1973

  23. [23]

    The American Naturalist , volume=

    Spiral chaos in a predator-prey model , author=. The American Naturalist , volume=. 1979 , publisher=

    work page 1979

  24. [24]

    Consequences of a refuge for the predator-prey dynamics of a wolf-elk system in

    Goldberg, Joshua F and Hebblewhite, Mark and Bardsley, John , journal=. Consequences of a refuge for the predator-prey dynamics of a wolf-elk system in. 2014 , publisher=

    work page 2014

  25. [25]

    Journal of Mathematical Biology , volume=

    Predator-prey populations with parasitic infection , author=. Journal of Mathematical Biology , volume=. 1989 , publisher=

    work page 1989

  26. [26]

    Predation, apparent competition, and the structure of prey communities , author=. Theo. Popu. Biol. , volume=. 1977 , publisher=

    work page 1977

  27. [27]

    Journal of Applied Mathematics and Computing , volume=

    Stability analysis of Filippov prey--predator model with fear effect and prey refuge , author=. Journal of Applied Mathematics and Computing , volume=. 2024 , publisher=

    work page 2024

  28. [28]

    1981 , publisher=

    Theory and applications of Hopf bifurcation , author=. 1981 , publisher=

    work page 1981

  29. [29]

    SIAM Review , volume=

    The mathematics of infectious diseases , author=. SIAM Review , volume=. 2000 , publisher=

    work page 2000

  30. [30]

    Theoretical Population Biology , volume=

    A predator--prey model with infected prey , author=. Theoretical Population Biology , volume=. 2004 , publisher=

    work page 2004

  31. [31]

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=

    Species competition: coexistence, exclusion and clustering , author=. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=. 2009 , publisher=

    work page 2009

  32. [32]

    Evolutionary Ecology , volume=

    Refuge evolution and the population dynamics of coupled host—parasitoid associations , author=. Evolutionary Ecology , volume=. 1995 , publisher=

    work page 1995

  33. [33]

    The Canadian Entomologist , volume=

    Some characteristics of simple types of predation and parasitism1 , author=. The Canadian Entomologist , volume=. 1959 , publisher=

    work page 1959

  34. [34]

    The Memoirs of the Entomological Society of Canada , volume=

    The functional response of predators to prey density and its role in mimicry and population regulation , author=. The Memoirs of the Entomological Society of Canada , volume=. 1965 , publisher=

    work page 1965

  35. [35]

    A theoretical framework for intraguild predation , author=. Am. Nat. , volume=. 1997 , publisher=

    work page 1997

  36. [36]

    Alternative prey and the dynamics of intraguild predation: theoretical perspectives , author=. Ecol. , volume=. 2007 , publisher=

    work page 2007

  37. [37]

    Chaos, Solitons & Fractals , volume=

    Impact of fear on an eco-epidemiological model , author=. Chaos, Solitons & Fractals , volume=. 2020 , publisher=

    work page 2020

  38. [38]

    Population Ecology , volume=

    Consequences of ratio-dependent predation by wolves for elk population dynamics , author=. Population Ecology , volume=. 2013 , publisher=

    work page 2013

  39. [39]

    International Journal of Bifurcation and Chaos , volume=

    Fear induced stabilization in an intraguild predation model , author=. International Journal of Bifurcation and Chaos , volume=. 2020 , publisher=

    work page 2020

  40. [40]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Bifurcations, chaos, and multistability in a nonautonomous predator--prey model with fear , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2021 , publisher=

    work page 2021

  41. [41]

    The effect of castration, of estrogen and of androgen injection on serum phosphatases in metastatic carcinoma of the prostate , author=

    Studies on prostatic cancer: I. The effect of castration, of estrogen and of androgen injection on serum phosphatases in metastatic carcinoma of the prostate , author=. The Journal of Urology , volume=. 2002 , publisher=

    work page 2002

  42. [42]

    Mathematical Biosciences and Engineering , volume=

    Global dynamics of a simple model for wild and sterile mosquitoes , author=. Mathematical Biosciences and Engineering , volume=

    work page

  43. [43]

    BMC Medical Genomics , volume=

    A Japanese case of castration-resistant prostate cancer with BRCA2 and RB1 co-loss and TP53 mutation: a case report , author=. BMC Medical Genomics , volume=. 2022 , publisher=

    work page 2022

  44. [44]

    Down To Earth , year =

    Shuchita Jha , title =. Down To Earth , year =

    work page

  45. [45]

    Oecologia , volume=

    Density-dependent effects of prey defences , author=. Oecologia , volume=. 2000 , publisher=

    work page 2000

  46. [46]

    Dynamics of a predator-prey system with prey subject to

    Kang, Yun and Sasmal, Sourav Kumar and Bhowmick, Amiya Ranjan and Chattopadhyay, Joydev , journal=. Dynamics of a predator-prey system with prey subject to. 2014 , publisher=

    work page 2014

  47. [47]

    Dynamics of a intraguild predation model with generalist or specialist predator , author=. J. Math. Biol. , volume=. 2013 , publisher=

    work page 2013

  48. [48]

    Communications in Nonlinear Science and Numerical Simulation , volume=

    Stability analysis of a prey--predator model incorporating a prey refuge , author=. Communications in Nonlinear Science and Numerical Simulation , volume=. 2005 , publisher=

    work page 2005

  49. [49]

    2001 , publisher=

    Elements of mathematical ecology , author=. 2001 , publisher=

    work page 2001

  50. [50]

    Science , volume=

    Foraging adaptation and the relationship between food-web complexity and stability , author=. Science , volume=. 2003 , publisher=

    work page 2003

  51. [51]

    Applied Mathematics and Computation , volume=

    Dynamic behavior of a Beddington--DeAngelis type stage structured predator--prey model , author=. Applied Mathematics and Computation , volume=. 2014 , publisher=

    work page 2014

  52. [52]

    Applied Mathematics and Computation , volume=

    Role of constant prey refuge on stage structure predator--prey model with ratio dependent functional response , author=. Applied Mathematics and Computation , volume=. 2017 , publisher=

    work page 2017

  53. [53]

    Mathematical Biosciences , volume=

    Uniqueness of limit cycles in Gause-type models of predator-prey systems , author=. Mathematical Biosciences , volume=. 1988 , publisher=

    work page 1988

  54. [54]

    1998 , publisher=

    Elements of applied bifurcation theory , author=. 1998 , publisher=

    work page 1998

  55. [55]

    Effects of optimal antipredator behavior of prey on predator--prey dynamics: the role of refuges , author=. Theo. Popu. Biol. , volume=. 1998 , publisher=

    work page 1998

  56. [56]

    Theoretical Population Biology , volume=

    Optimal foraging and predator--prey dynamics, II , author=. Theoretical Population Biology , volume=. 1999 , publisher=

    work page 1999

  57. [57]

    Theoretical Population Biology , volume=

    Optimal foraging and predator--prey dynamics , author=. Theoretical Population Biology , volume=. 1996 , publisher=

    work page 1996

  58. [58]

    Theoretical Population Biology , volume=

    Optimal intraguild foraging and population stability , author=. Theoretical Population Biology , volume=. 2000 , publisher=

    work page 2000

  59. [59]

    Theoretical Population Biology , volume=

    Adaptive omnivory and species coexistence in tri-trophic food webs , author=. Theoretical Population Biology , volume=. 2005 , publisher=

    work page 2005

  60. [60]

    Trait adaptation enhances species coexistence and reduces bistability in an intraguild predation module , author=. Ecol. Evol. , volume=. 2023 , publisher=

    work page 2023

  61. [61]

    Canadian Journal of Zoology , volume=

    Behavioral decisions made under the risk of predation: a review and prospectus , author=. Canadian Journal of Zoology , volume=. 1990 , publisher=

    work page 1990

  62. [62]

    Animal Behaviour , volume=

    Dangerous prey is associated with a type 4 functional response in spiders , author=. Animal Behaviour , volume=. 2013 , publisher=

    work page 2013

  63. [63]

    Criterion of Hopf bifurcations without using eigenvalues , author=. J. Math. Anal. Appl. , volume=. 1994 , publisher=

    work page 1994

  64. [64]

    Williams and Wilkins , year=

    Elements of physical biology , author=. Williams and Wilkins , year=

    work page

  65. [65]

    Applied Mathematics and Computation , volume=

    Persistence and extinction in an Elk-Wolf prey-predator system with refuge and inter-regional movement , author=. Applied Mathematics and Computation , volume=. 2026 , publisher=

    work page 2026

  66. [66]

    , author=

    Regulation of top-down effect due to anti-predator behavior of intraguild-prey in an intraguild predation model. , author=. Nonl. Dyn. , volume=. 2025 , publisher=

    work page 2025

  67. [67]

    Godwin, M

    An essay on the principle of population, as it affects the future improvement of society, with remarks on the speculations of Mr. Godwin, M. Condorcet, and other writers , author=. Charity and Philanthropy in Nineteenth-Century Britain , pages=. 1798 , publisher=

    work page

  68. [68]

    2015 , publisher=

    An introduction to mathematical epidemiology , author=. 2015 , publisher=

    work page 2015

  69. [69]

    Journal of Theoretical Biology , volume=

    A methodology for performing global uncertainty and sensitivity analysis in systems biology , author=. Journal of Theoretical Biology , volume=. 2008 , publisher=

    work page 2008

  70. [70]

    1974 , publisher=

    Models in ecology , author=. 1974 , publisher=

    work page 1974

  71. [71]

    Mathematical Biosciences , volume=

    The effect of prey refuge in a patchy predator--prey system , author=. Mathematical Biosciences , volume=. 2013 , publisher=

    work page 2013

  72. [72]

    2002 , publisher=

    Parasites and the behavior of animals , author=. 2002 , publisher=

    work page 2002

  73. [73]

    Journal of the Royal Society Interface , volume=

    Predators indirectly control vector-borne disease: linking predator--prey and host--pathogen models , author=. Journal of the Royal Society Interface , volume=. 2010 , publisher=

    work page 2010

  74. [74]

    Nonlinear Dynamics , volume=

    Impact of fear on a predator--prey system with prey-dependent search rate in deterministic and stochastic environment , author=. Nonlinear Dynamics , volume=. 2021 , publisher=

    work page 2021

  75. [75]

    Mathematics and Computers in Simulation , volume=

    Can adaptive prey refuge facilitate species coexistence in Bazykin’s prey--predator model? , author=. Mathematics and Computers in Simulation , volume=. 2025 , publisher=

    work page 2025

  76. [76]

    Applied Mathematics and Computation , volume=

    Hopf bifurcation in an eco-epidemic model , author=. Applied Mathematics and Computation , volume=. 2010 , publisher=

    work page 2010

  77. [77]

    The American Naturalist , volume=

    Refuge dynamics and metapopulation dynamics: an experimental test , author=. The American Naturalist , volume=. 1996 , publisher=

    work page 1996

  78. [78]

    Ecological Monographs , volume=

    Switching in general predators: experiments on predator specificity and stability of prey populations , author=. Ecological Monographs , volume=. 1969 , publisher=

    work page 1969

  79. [79]

    Oikos , volume=

    Influence of predator-specific defense adaptation on intraguild predation , author=. Oikos , volume=. 2010 , publisher=

    work page 2010

  80. [80]

    The American Naturalist , volume=

    Food web complexity and species diversity , author=. The American Naturalist , volume=. 1966 , publisher=

    work page 1966

Showing first 80 references.