A prey-predator model with three interacting species

Isabell Vorkastner; Michael Scheutzow; Uygun Jamilov

arxiv: 1907.05100 · v1 · pith:OLCBX6IAnew · submitted 2019-07-11 · 🧮 math.DS

A prey-predator model with three interacting species

Uygun Jamilov , Michael Scheutzow , Isabell Vorkastner This is my paper

Pith reviewed 2026-05-24 23:13 UTC · model grok-4.3

classification 🧮 math.DS

keywords prey-predator modeldiscrete time dynamicssimplexergodic hypothesisCesaro meansfixed pointpopulation dynamicsspeed function

0 comments

The pith

In this discrete prey-predator model on the simplex, parameter choices split the long-term behavior into diverging Cesaro means, attraction to the interior fixed point, or attraction to a vertex.

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

The paper studies a family of discrete-time maps on the two-dimensional simplex that track relative frequencies of three interacting species. For one class of parameters the time averages of any order are shown to diverge, so the ergodic hypothesis fails. For a second class every interior orbit converges to the single interior fixed point. For the remaining parameters every orbit converges to one of the three vertices. A speed function f is included so that small values of f make the discrete steps approximate continuous-time flows arbitrarily closely. Readers might care because the results identify when discrete population models lose the averaging properties expected from their continuous counterparts.

Core claim

The evolution operator on the simplex, defined with an arbitrary speed function f, produces three qualitatively distinct regimes according to parameter choice: for some parameters any Cesaro mean of trajectories diverges, for others all interior orbits converge to the unique fixed point, and for the rest all orbits converge to a vertex of the simplex. These statements are established for particular families of the operator.

What carries the argument

The discrete evolution operator on the two-dimensional simplex that updates relative frequencies of three species and incorporates an arbitrary speed function f.

If this is right

When parameters place the operator in the first class, the failure of the ergodic hypothesis means long-term frequency averages do not approach any stationary distribution.
When parameters place the operator in the second class, species frequencies stabilize at an interior equilibrium from any interior starting point.
When parameters place the operator in the third class, one species frequency reaches 1 while the others reach 0, regardless of interior starting point.
Small values of the speed function f make the discrete trajectories approximate the corresponding continuous-time flow arbitrarily closely.

Where Pith is reading between the lines

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

The split into three regimes suggests that discrete-time ecological models can exhibit extinction or non-averaging behavior even when their continuous limits do not.
One could compare the discrete orbits for small f against numerical solutions of the associated ODE system to quantify how well the speed function approximates continuous dynamics.
The results indicate that coexistence equilibria in such models are structurally fragile with respect to the choice of discrete step size.

Load-bearing premise

The convergence and divergence statements are proved only for particular families of the operator that incorporates the speed function f.

What would settle it

A numerical iteration of the operator for a parameter set asserted to produce diverging Cesaro means that instead produces bounded averages of some order would falsify the divergence claim.

read the original abstract

In this paper we consider a class of discrete time prey-predator models with three interacting species defined on the two-dimensional simplex. For some choices of parameters of the operator describing the evolution of the relative frequencies, we show that the ergodic hypothesis does not hold. Moreover, we prove that any order Ces\`aro mean of the trajectories diverges. For another class of parameters, we show that all orbits starting from the interior of the simplex converge to the unique fixed point of the operator while for the remaining choices of parameters all orbits converge to one of the vertices of the simplex. Contrary to many authors we study discrete time models but we include a speed function $f$ in the dynamics which allows us to approximate the continuous-time case arbitrarily well when $f$ is small.

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.

Desk Editor's Note private letter to a colleague

The paper proves a trichotomy of orbit behaviors for a parameterized family of three-species maps on the simplex, with explicit cases of Cesaro divergence, but the results hold only for restricted families of the speed function f.

read the letter

The core contribution is a clean partition of parameter space for this discrete map: some regimes produce diverging Cesaro means so the ergodic hypothesis fails, others send all interior orbits to the unique fixed point, and the rest send them to vertices. The speed function f is included to let the discrete step approximate continuous-time dynamics when f is small. That combination of tunable f with explicit regime statements looks new compared to the cited prior work on similar maps. The analysis is direct on the iteration operator and avoids circularity or fitted parameters. The proofs are claimed to exist for the stated behaviors, which is a plus for a pure math paper in this area. The main limitation is that the trichotomy is established only for particular families of f (constant or linear, for example), not for arbitrary positive speed functions. The abstract presents the model with general f, yet the derivations apparently rely on extra structure that is not justified in general. If a counter-example f exists outside those families, the claimed exhaustive partition does not cover the full model as stated. No data or external validation is involved, so the work stands or falls on the correctness of those derivations. This is a modest technical result useful to people who already work on discrete population models and want precise convergence statements for low-dimensional simplex maps. It is not a reorganization of the broader field. The paper shows clear engagement with the literature and direct reasoning on its own terms, so it merits a serious referee even if revisions are needed to clarify the scope of f.

Referee Report

1 major / 0 minor

Summary. The paper considers a discrete-time prey-predator model with three species on the 2-simplex, governed by an iteration operator that includes a positive speed function f. It claims a trichotomy on long-term behavior depending on parameters: for some parameter values the ergodic hypothesis fails and Cesàro means of all orders diverge; for another class all interior orbits converge to the unique fixed point; for the remaining parameters all orbits converge to a vertex. The inclusion of f is presented as allowing arbitrary approximation of the continuous-time limit.

Significance. A rigorous trichotomy for the general model would be a substantive contribution to discrete dynamical systems in population biology, providing explicit counter-examples to ergodicity and a complete classification of attractors. The limitation of the proofs to particular families of f (rather than arbitrary positive f) substantially reduces the scope and strength of the result.

major comments (1)

[Abstract] Abstract and §1: the trichotomy is stated for the general operator with arbitrary positive speed function f, yet the derivations establishing divergence of Cesàro means, convergence to the fixed point, and convergence to vertices are supplied only for specific families of f (constant or linear). This is load-bearing for the central claim of an exhaustive partition of parameter space.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the important issue of the scope of the trichotomy with respect to the speed function f. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and §1: the trichotomy is stated for the general operator with arbitrary positive speed function f, yet the derivations establishing divergence of Cesàro means, convergence to the fixed point, and convergence to vertices are supplied only for specific families of f (constant or linear). This is load-bearing for the central claim of an exhaustive partition of parameter space.

Authors: We agree that the rigorous proofs of the three cases in the trichotomy (divergence of all Cesàro means, convergence of interior orbits to the interior fixed point, and convergence to vertices) are carried out only for the constant and linear families of f. The abstract and introduction describe the model with a general positive speed function f, but the theorems themselves are proved under these restrictions. We will revise the abstract and §1 to state explicitly that the trichotomy holds for constant and linear speed functions f. This change will align the claims with the proved results while retaining the paper's contribution of explicit counter-examples to ergodicity and a complete classification for these families. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proofs on explicitly defined operator

full rationale

The paper defines an explicit discrete-time operator on the simplex that includes a general positive speed function f and then proves orbit behavior (Cesàro divergence, convergence to fixed point, or convergence to vertices) for partitioned parameter classes via direct analysis. No fitted parameters are renamed as predictions, no self-definitional loops appear, and no load-bearing self-citations or imported uniqueness theorems are invoked. The derivation chain consists of standard dynamical-systems arguments applied to the stated model and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from discrete dynamical systems on simplices and the definition of Cesaro means; no new entities are postulated and no parameters are fitted to data.

axioms (1)

standard math The state space is the standard 2-simplex equipped with the usual topology and iteration of a continuous map.
The model is defined directly on this space; all convergence statements presuppose the standard properties of compact metric spaces and continuous functions.

pith-pipeline@v0.9.0 · 5661 in / 1312 out tokens · 43914 ms · 2026-05-24T23:13:07.048477+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.8. If a>0,b>0,c>0, then for each k the vertices e1,e2,e3 are limit points of the k-th order Cesaro sequences... W is a non-ergodic transformation.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 2.9. The function ϕ(x)=x^λ1_1 x^λ2_2 x^λ3_3 is a Lyapunov function... ω(x(0)) is an infinite subset of ∂S^2

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

22 extracted references · 22 canonical work pages

[1]

E. Akin, V. Losert, Evolutionary dynamics of zero-sum ga mes, J. Math. Biology 20 (1984) 231–258

work page 1984
[2]

Bernstein, The solution of a mathematical problem rel ated to the theory of heredity, Uchn

S. Bernstein, The solution of a mathematical problem rel ated to the theory of heredity, Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat. 1 (1924) 83–115

work page 1924
[3]

Chauvet, J

E. Chauvet, J. E. Paullet, J. P. Privite, Z. Walls, A Lotka -Volterra three-species food chain, Math. Magazine. 75 (4) (2002) 243–255

work page 2002
[4]

R Davronov, U

R. R Davronov, U. U. Jamilov (Zhamilov), M. Ladra, Condit ional cubic stochastic operator, Jour. Diﬀ. Equ. Appl. 21 (12) (2015) 1163–1170

work page 2015
[5]

H. I. Freedman, P. Waltman, Persistence in models of thre e interacting predator-prey populations, Math. Biosci. 68 (2) (1984) 213–231

work page 1984
[6]

H. I. Freedman, P. Waltman, Mathematical analysis of som e three-species food-chain models, Math. Biosci. 33 (3-4) (1977) 257–276

work page 1977
[7]

N. N. Ganikhodzhaev, D. V. Zanin, On a necessary conditio n for the ergodicity of quadratic operators deﬁned on a two-dimensional simplex, Russ. Math. Surv. 59 (3 ) (2004) 571–572

work page 2004
[8]

Ganikhodjaev, R

N. Ganikhodjaev, R. Ganikhodjaev, U. Jamilov, Quadrati c stochastic operators and zero-sum game dynamics, Ergodic Theory Dynam. Systems 35(5) (2015) 1443– 1473

work page 2015
[9]

Ganikhodzhaev, F

R. Ganikhodzhaev, F. Mukhamedov, U. Rozikov, Quadratic stochastic operators and processes: results and open problems, Inﬁn. Dimens. Anal. Quantum Prob ab. Relat. Top. 14 (2) (2011) 279– 335

work page 2011
[10]

T. C. Gard, T. G. Hallam, Persistence in food webs. I. Lot ka-Volterra food chains, Bull. Math. Biol. 41 (6) (1979) 877–891

work page 1979
[11]

A. J. Homburg, U. U. Jamilov, M. Scheutzow, Asymptotics for a class of iterated random cubic operators, Nonlinearity (to appear)

work page
[12]

U. U. Jamilov, A. Yu. Khamraev, M. Ladra, On a Volterra cu bic stochastic operator, Bull. Math. Biol. 80 (2) (2018) 319–334

work page 2018
[13]

U. U. Jamilov, M. Ladra, On identically distributed non -Volterra cubic stochastic operator, J. Appl. Nonlinear Dyn. 6 (1) (2017) 79–90

work page 2017
[14]

U. U. Jamilov, M. Scheutzow, I. Vorkastner, Strong pers istence of a discrete-time population model with three species, http://page.math.tu-berlin.de/∼vorkastn/stability_LV_model.pdf, Preprint (2019)

work page 2019
[15]

Kesten, Quadratic transformations: A model for popu lation growth

H. Kesten, Quadratic transformations: A model for popu lation growth. I, Advances in Appl. Prob- ability 2 (1970) 1–82

work page 1970
[16]

Y. I. Lyubich, Mathematical structures in population g enetics, vol. 22 of Biomathematics, Springer- Verlag, Berlin, 1992

work page 1992
[17]

J. D. Murray, Mathematical biology. I. An introduction . Third edition. Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002

work page 2002
[18]

U. A. Rozikov, A. Yu. Khamraev, On cubic operators deﬁne d on ﬁnite-dimensional simpleces. Ukraine Math. Jour. 56 (10) (2004) 1424–1433

work page 2004
[19]

U. A. Rozikov, A. Yu. Khamraev, On construction and a cla ss of non-Volterra cubic stochastic operators, Nonlinear Dynamics and System Theory. 14 (1) (20 14) 92–100

work page
[20]

Saburov, A class of nonergodic Lotka-Volterra opera tors, Math

M. Saburov, A class of nonergodic Lotka-Volterra opera tors, Math. Notes. 97 (5) (2015) 759–763

work page 2015
[21]

S. M. Ulam, A collection of mathematical problems, Inte rscience Tracts in Pure and Applied Math- ematics, no. 8, Interscience Publishers, New York-London, 1960

work page 1960
[22]

M. I. Zakharevich, On the behaviour of trajectories and the ergodic hypothesis for quadratic map- pings of a simplex, Russ. Math. Surv. 33 (6) (1978) 265–266. U. U. Jamilov, V.I. Romanovskiy Institute of Ma thema tics, U zbekistan Academy of Sciences, 81, Mirzo-Ulugbek str., 100170, Tashkent, Uzbek istan. E-mail address : jamilovu@yandex.ru 14 U.U. JAMILO...

work page 1978

[1] [1]

E. Akin, V. Losert, Evolutionary dynamics of zero-sum ga mes, J. Math. Biology 20 (1984) 231–258

work page 1984

[2] [2]

Bernstein, The solution of a mathematical problem rel ated to the theory of heredity, Uchn

S. Bernstein, The solution of a mathematical problem rel ated to the theory of heredity, Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat. 1 (1924) 83–115

work page 1924

[3] [3]

Chauvet, J

E. Chauvet, J. E. Paullet, J. P. Privite, Z. Walls, A Lotka -Volterra three-species food chain, Math. Magazine. 75 (4) (2002) 243–255

work page 2002

[4] [4]

R Davronov, U

R. R Davronov, U. U. Jamilov (Zhamilov), M. Ladra, Condit ional cubic stochastic operator, Jour. Diﬀ. Equ. Appl. 21 (12) (2015) 1163–1170

work page 2015

[5] [5]

H. I. Freedman, P. Waltman, Persistence in models of thre e interacting predator-prey populations, Math. Biosci. 68 (2) (1984) 213–231

work page 1984

[6] [6]

H. I. Freedman, P. Waltman, Mathematical analysis of som e three-species food-chain models, Math. Biosci. 33 (3-4) (1977) 257–276

work page 1977

[7] [7]

N. N. Ganikhodzhaev, D. V. Zanin, On a necessary conditio n for the ergodicity of quadratic operators deﬁned on a two-dimensional simplex, Russ. Math. Surv. 59 (3 ) (2004) 571–572

work page 2004

[8] [8]

Ganikhodjaev, R

N. Ganikhodjaev, R. Ganikhodjaev, U. Jamilov, Quadrati c stochastic operators and zero-sum game dynamics, Ergodic Theory Dynam. Systems 35(5) (2015) 1443– 1473

work page 2015

[9] [9]

Ganikhodzhaev, F

R. Ganikhodzhaev, F. Mukhamedov, U. Rozikov, Quadratic stochastic operators and processes: results and open problems, Inﬁn. Dimens. Anal. Quantum Prob ab. Relat. Top. 14 (2) (2011) 279– 335

work page 2011

[10] [10]

T. C. Gard, T. G. Hallam, Persistence in food webs. I. Lot ka-Volterra food chains, Bull. Math. Biol. 41 (6) (1979) 877–891

work page 1979

[11] [11]

A. J. Homburg, U. U. Jamilov, M. Scheutzow, Asymptotics for a class of iterated random cubic operators, Nonlinearity (to appear)

work page

[12] [12]

U. U. Jamilov, A. Yu. Khamraev, M. Ladra, On a Volterra cu bic stochastic operator, Bull. Math. Biol. 80 (2) (2018) 319–334

work page 2018

[13] [13]

U. U. Jamilov, M. Ladra, On identically distributed non -Volterra cubic stochastic operator, J. Appl. Nonlinear Dyn. 6 (1) (2017) 79–90

work page 2017

[14] [14]

U. U. Jamilov, M. Scheutzow, I. Vorkastner, Strong pers istence of a discrete-time population model with three species, http://page.math.tu-berlin.de/∼vorkastn/stability_LV_model.pdf, Preprint (2019)

work page 2019

[15] [15]

Kesten, Quadratic transformations: A model for popu lation growth

H. Kesten, Quadratic transformations: A model for popu lation growth. I, Advances in Appl. Prob- ability 2 (1970) 1–82

work page 1970

[16] [16]

Y. I. Lyubich, Mathematical structures in population g enetics, vol. 22 of Biomathematics, Springer- Verlag, Berlin, 1992

work page 1992

[17] [17]

J. D. Murray, Mathematical biology. I. An introduction . Third edition. Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002

work page 2002

[18] [18]

U. A. Rozikov, A. Yu. Khamraev, On cubic operators deﬁne d on ﬁnite-dimensional simpleces. Ukraine Math. Jour. 56 (10) (2004) 1424–1433

work page 2004

[19] [19]

U. A. Rozikov, A. Yu. Khamraev, On construction and a cla ss of non-Volterra cubic stochastic operators, Nonlinear Dynamics and System Theory. 14 (1) (20 14) 92–100

work page

[20] [20]

Saburov, A class of nonergodic Lotka-Volterra opera tors, Math

M. Saburov, A class of nonergodic Lotka-Volterra opera tors, Math. Notes. 97 (5) (2015) 759–763

work page 2015

[21] [21]

S. M. Ulam, A collection of mathematical problems, Inte rscience Tracts in Pure and Applied Math- ematics, no. 8, Interscience Publishers, New York-London, 1960

work page 1960

[22] [22]

M. I. Zakharevich, On the behaviour of trajectories and the ergodic hypothesis for quadratic map- pings of a simplex, Russ. Math. Surv. 33 (6) (1978) 265–266. U. U. Jamilov, V.I. Romanovskiy Institute of Ma thema tics, U zbekistan Academy of Sciences, 81, Mirzo-Ulugbek str., 100170, Tashkent, Uzbek istan. E-mail address : jamilovu@yandex.ru 14 U.U. JAMILO...

work page 1978