pith. sign in

arxiv: 1907.06079 · v1 · pith:QFKTDHTLnew · submitted 2019-07-13 · 🧮 math.DS · q-bio.PE

Large and Small Data Blow-Up Solutions in the Trojan Y Chromosome Model

Rana D. Parshad , Matthew A. Beauregard , Eric M. Takyi , Thomas Griffin , Landrey Bobo This is my paper

Pith reviewed 2026-05-24 21:49 UTC · model grok-4.3

classification 🧮 math.DS q-bio.PE
keywords trojan y chromosomeblow-up solutionsnegative solutionsfinite time blow-upinvasive species controlreaction-diffusion modelsex ratio dynamics
0
0 comments X

The pith

The TYC model permits negative male solutions that produce finite-time blow-up when trojan fish introduction rate is zero under large data or large enough under any positive data.

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

The paper examines the Trojan Y Chromosome model without assuming solutions stay positive. It shows that zero trojan introduction combined with large initial data allows the male population to become negative, after which both female and male populations blow up in finite time. The same blow-up occurs for every positive initial condition once the introduction rate is high enough. Parallel blow-up results hold in the reaction-diffusion version of the model. The work also explores possible changes to the equations that might eliminate negativity or slow the blow-up.

Core claim

If the introduction rate of trojan fish is zero, under certain large data assumptions, negative solutions are possible for the male population, which in turn can lead to finite time blow-up in the female and male populations. A comparable result is established for any positive initial condition if the introduction rate of trojan fish is large enough. Similar finite time blow-up results are obtained in a spatial temporal TYC model that includes diffusion.

What carries the argument

The TYC system of ODEs (and its reaction-diffusion extension) for female, male, and trojan populations, whose right-hand sides permit sign changes that drive blow-up.

Load-bearing premise

The TYC model equations remain valid and meaningful even when solutions become negative.

What would settle it

A numerical integration of the TYC system under the stated large-data or high-rate conditions that stays non-negative and bounded for all time would falsify the blow-up result.

Figures

Figures reproduced from arXiv: 1907.06079 by Eric M. Takyi, Landrey Bobo, Matthew A. Beauregard, Rana D. Parshad, Thomas Griffin.

Figure 1
Figure 1. Figure 1: The pedigree tree of the TYC model (that demonstrates Trojan Y-Chromosome eradication strategy). (a) Mating of a wild-type XX female (f) and a wild-type XY male (m). (b) Mating of a wild￾type XY male (m) and a sex-reversed YY female (r). (c) Mating of a wild-type XX female (f) and a YY supermale (s). (d) Mating of a sex-reversed YY trojan female (r) and a YY supermale (s). Red color represents wild types, … view at source ↗
Figure 2
Figure 2. Figure 2: (a) Positive solutions are shown given the initial condi￾tions of f(0) = m(0) = .3 and s(0) = .1. (b) The simulation with f(0) = m(0) = .3 and s(0) = 2.5. Notice that the male population is clearly negative for an interval. Hence, s(0) = 2.5 > s∗ . (c) The population densities given initial conditions of f(0) = m(0) = .4 and s(0) = 2.5. The female population is tending towards infinity and we estimate the … view at source ↗
Figure 3
Figure 3. Figure 3: Three regions in the phase space are shown. In Region 1, where s(0) < s∗ , positive solutions are guaranteed. In Region 2, where s ∗∗ > s(0) ≥ s ∗ , negative solutions exist but finite time blow up does not occur in either the female or male populations. In Region 3, for s(0) ≥ s ∗∗ negative solutions exist and blow-up in finite time occurs. lim supt→T ∗<∞ f → +∞ for finite time T ∗ , deemed the blow-up ti… view at source ↗
Figure 4
Figure 4. Figure 4: Three regions in the phase space are shown. In Region 1, where γ < γ∗ , positive solutions are guaranteed. In Region 2, where γ ∗∗ > γ ≥ γ ∗ , negative solutions exist but finite time blow up does not occur in either the female or male populations. In Region 3, for γ ≥ γ ∗∗ negative solutions exist and blow-up in finite time occurs. In each simulation, s(0) = 0. It is evident that both thresholds are indep… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Simulation showing negative solutions in the male population for initial conditions f(x, 0) = m(x, 0) = .3 and s(x, 0) = 2.5. (b) Simulation with f(x, 0) = m(x, 0) = .3 and s(x, 0) = 2.75. The increase in the initial amount of supermales results in finite time blow-up in the female population. Simulations were conducted used Matlabr’s pdepe built-in partial differential solver. 5. Discussion and Conclu… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Simulation showing negative solutions in the male population for initial conditions f(x, 0) = m(x, 0) = x(1 − x) and s(x, 0) = 4smaxx(1 − x), where smax = 2. (b) Simulation with the same initial conditions but with smax = 3. The increase in the maximum number supermales results in finite time blow-up in the female population. Each simulation uses D = .01. as in (18)-(20) is: ˙f = rL  f a − 1   m m +… view at source ↗
Figure 7
Figure 7. Figure 7: Two regions in the phase space are shown. In Region 1, where s(0) < s∗ , positive solutions are guaranteed. In Region 2, where s(0) ≥ s ∗ , negative solutions exist. No simulation rendered finite time blow-up in either the female or male populations. In contrast, the Allee effect affects the differentiability of the threshold curve since at small populations the Allee effect can dominate the dynamics. To i… view at source ↗
Figure 8
Figure 8. Figure 8: Two regions in the phase space are shown. In Region 1, where s(0) < s∗ , positive solutions are guaranteed. In Region 2, where s(0) ≥ s ∗ , negative solutions exist. No simulation rendered finite time blow-up in either the female or male populations. The threshold for (red) Eqs. (18)-(20) and (blue) Eqs. (24)-(26) with no Allee effect are given. Alternatives to the logistic term are plentiful. For instance… view at source ↗
read the original abstract

The Trojan Y Chromosome Strategy (TYC) is an extremely well investigated biological control method for controlling invasive populations with an XX-XY sex determinism. In \cite{GP12, WP14} various dynamical properties of the system are analyzed, including well posedness, boundedness of solutions, and conditions for extinction or recovery. These results are derived under the assumption of positive solutions. In the current manuscript, we show that if the introduction rate of trojan fish is zero, under certain large data assumptions, negative solutions are possible for the male population, which in turn can lead to finite time blow-up in the female and male populations. A comparable result is established for \emph{any} positive initial condition if the introduction rate of trojan fish is large enough. Similar finite time blow-up results are obtained in a spatial temporal TYC model that includes diffusion. Lastly, we investigate improvements to the TYC modeling construct that may dampen the mechanisms to the blow-up phenomenon or remove the negativity of solutions. The results draw into suspect the reliability of current TYC models under certain situations.

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 manuscript analyzes the Trojan Y Chromosome (TYC) ODE and PDE models. It shows that with zero trojan introduction rate and suitable large-data initial conditions the male component can become negative, after which both male and female components exhibit finite-time blow-up; an analogous statement holds for sufficiently large positive introduction rates and arbitrary positive initial data. Parallel blow-up results are obtained for the diffusive spatial-temporal version. The authors also examine model modifications intended to suppress negativity or blow-up and conclude that existing TYC models are unreliable under the identified regimes.

Significance. If the derivations are correct, the explicit construction of negative solutions and the resulting blow-up provides a concrete mathematical counter-example to the positivity hypotheses used in the cited prior works (GP12, WP14). This supplies a precise limitation on the range of validity of those earlier boundedness and extinction results. The parallel PDE analysis and the discussion of possible modeling fixes add technical value to the critique of the TYC construct.

major comments (3)
  1. [ODE blow-up theorem (likely §3)] The central ODE blow-up claim (zero-rate case) rests on the assertion that negativity of the male component forces finite-time blow-up in the coupled system. The manuscript should supply the precise differential inequality or comparison argument used after the male variable crosses zero; without it the transition from negativity to blow-up remains a black box.
  2. [Large-rate theorem (likely §3)] For the large-introduction-rate result that holds for arbitrary positive data, the threshold on the introduction parameter must be shown to be independent of the initial data size; otherwise the statement reduces to a restatement of the large-data case already treated.
  3. [PDE section (likely §4)] In the diffusive PDE version the manuscript asserts analogous finite-time blow-up. Because diffusion is typically smoothing, the proof must identify whether blow-up still occurs for any positive diffusion coefficient or only in the zero-diffusion limit; the current statement leaves this distinction unclear.
minor comments (2)
  1. Notation for the introduction rate parameter is used inconsistently between the zero-rate and positive-rate statements; a single symbol with an explicit range would improve readability.
  2. [Introduction] The abstract states that the results 'draw into suspect the reliability' of current models; the introduction should cite the exact positivity hypotheses from GP12 and WP14 so readers can see the precise contrast.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify several points in the manuscript. We address each major comment below and will revise accordingly to improve the presentation.

read point-by-point responses

  1. Referee: [ODE blow-up theorem (likely §3)] The central ODE blow-up claim (zero-rate case) rests on the assertion that negativity of the male component forces finite-time blow-up in the coupled system. The manuscript should supply the precise differential inequality or comparison argument used after the male variable crosses zero; without it the transition from negativity to blow-up remains a black box.

    Authors: We agree that the transition step requires more explicit detail. After the male component M(t) becomes negative at some finite time t*, substitution into the female equation yields F' ≥ (r/K) F |M| minus lower-order terms. Since |M| remains bounded away from zero on a short interval after t* (by continuity), this produces the superlinear inequality F' ≥ c F^2 for a positive constant c, which is compared directly to the explicitly solvable ODE v' = c v^2 that blows up in finite time. We will insert this differential inequality and the comparison argument verbatim in the revised proof of the zero-rate theorem. revision: yes

  2. Referee: [Large-rate theorem (likely §3)] For the large-introduction-rate result that holds for arbitrary positive data, the threshold on the introduction parameter must be shown to be independent of the initial data size; otherwise the statement reduces to a restatement of the large-data case already treated.

    Authors: The threshold β* on the trojan introduction rate is constructed explicitly from the model parameters r, d, and K alone and does not depend on the size of the initial data. The proof obtains an upper bound on the time to negativity that is uniform in the initial data by using the most adverse (largest) possible growth estimates; once negativity occurs, the same blow-up argument as in the zero-rate case applies. We will add an explicit remark and a short calculation confirming independence from initial data in the revised version of the large-rate theorem. revision: yes

  3. Referee: [PDE section (likely §4)] In the diffusive PDE version the manuscript asserts analogous finite-time blow-up. Because diffusion is typically smoothing, the proof must identify whether blow-up still occurs for any positive diffusion coefficient or only in the zero-diffusion limit; the current statement leaves this distinction unclear.

    Authors: The blow-up result holds for every fixed diffusion coefficient D > 0. Integrating the PDE system over the spatial domain (with no-flux boundary conditions) causes all diffusion terms to vanish, so the total populations satisfy exactly the same ODE system analyzed in Section 3. Consequently the integrated quantities blow up in finite time independently of D, which forces the PDE solution itself to become unbounded in L^∞. We will state this reduction explicitly and note the independence from D in the revised PDE section. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper conducts a direct mathematical analysis of the TYC ODE and PDE systems taken from the cited prior works, proving that negativity of the male component is possible under zero or sufficiently large trojan introduction rates and that this negativity triggers finite-time blow-up. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; the blow-up statements follow from standard comparison principles and integration of the explicit ODE right-hand sides under the stated large-data or high-rate hypotheses. The contrast with positivity assumptions in the references is presented as an external observation rather than an internal derivation that loops back on itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theoretical analysis extending prior TYC dynamical system models. It relies on the standard form of the population equations from cited works and standard techniques for proving blow-up in ODE/PDE systems. No new entities are postulated.

axioms (1)
  • domain assumption The TYC model equations as defined in GP12 and WP14
    The analysis builds directly on the system analyzed in the cited prior works for positive solutions.

pith-pipeline@v0.9.0 · 5739 in / 1319 out tokens · 31010 ms · 2026-05-24T21:49:03.252760+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    M. Arim, S. Abades, P. Neill, M. Lima and P. Marquet, Spread Dynamics of invasive species, P. Natl. Acad. Sci. USA, Vol. 103, No. 2, pp. 374-378, 2006

    work page 2006

  2. [2]

    Averill and Y.Lou, On several conjectures from evolution of dispersal , J

    I. Averill and Y.Lou, On several conjectures from evolution of dispersal , J. Biol. Dynam., Vol. 6, No. 2, pp. 117-130, 2012

    work page 2012

  3. [3]

    Bampfylde and M.A

    C.J. Bampfylde and M.A. Lewis, Biological control through intraguild predation:case studies in pest control, invasive species and range expansion , B. Math. Biol., Vol. 69, pp. 1031-1066, 2007

    work page 2007

  4. [4]

    Optimal Control and Analysis of a Modified Trojan Y-Chromosome Strategy

    M.A. Beauregard, R.D. Parshad, S. Boon, H. Conaway, T. Griffin, and J. Lyu, Optimal Control and Analysis of a Modified Trojan Y-Chromosome Strategy , arXiv:1907.03818

    work page internal anchor Pith review Pith/arXiv arXiv 1907

  5. [5]

    Castillo-Chavez, W

    C. Castillo-Chavez, W. Huang, The Logistic Equation Revisited: The Two-Sex Case Math. Biosci., Vol. 128, pp. 299-316, 1995

    work page 1995

  6. [6]

    Clark, M

    J.S. Clark, M. Lewis and L. Horvath, Invasion by extremes: Population spread with variation in dispersal and reproduction, Am. Nat., Vol. 157, No. 5, 2001

    work page 2001

  7. [7]

    Cotton and C

    S. Cotton and C. Wedekind Control of introduced species using Trojan sex chromosomes , Trends Ecol. Evol., Vol. 22, pp. 441-443, 2007

    work page 2007

  8. [8]

    Cotton and C

    S. Cotton and C. Wedekind, Introduction of Trojan sex chromosomes to boost population growth, J. Theor. Biol., Vol. 249, pp. 153-161, 2007

    work page 2007

  9. [9]

    Cotton and C

    S. Cotton and C. Wedekind, Population consequences of environmental sex reversal, Conserv. Biol., Vol. 23, pp. 196-206, 2009

    work page 2009

  10. [10]

    Gutierrez and J

    J.B. Gutierrez and J. Teem A model describing the effect of sex-reversed YY fish in an established wild population: the use of a Trojan Y-Chromosome to cause extinction of an introduced exotic species, J. Theor. Biol, Vol. 241, No. 22, pp. 333-341, 2006

    work page 2006

  11. [11]

    J. B. Gutierrez, M. K. Hurdal, R. D. Parshad and J. L. Teem, Analysis of the Trojan Y Chromosome Model for Eradication of Invasive Species in a Dendritic Riverine System , J. Math. Biol., Vol. 64, No. 1-2, pp. 319-340, 2012

    work page 2012

  12. [12]

    Hadeler, Pair Formation in Age-Structured Populations, Acta

    K.P. Hadeler, Pair Formation in Age-Structured Populations, Acta. Appl. Math., Vol 14, pp 91-102, 1989

    work page 1989

  13. [13]

    Kennedy, K.A

    P.A. Kennedy, K.A. Meyer, D.J. Schill, M. R. Campbell, and N.V. Vue, Survival and re- production success of hatchery YY male brook trout stocked in Idaho streams , T. Am. Fish. Soc., Vol. 147, pp. 419-430, 2018

    work page 2018

  14. [14]

    Kramer, The evidence for Allee effects , Popul

    A.M. Kramer, The evidence for Allee effects , Popul. Ecol., Vol. 51, pp. 341354 (2009)

    work page 2009

  15. [15]

    Lou and D

    Y. Lou and D. Munther, Dynamics of a three species competition model , Discret. Contin. Dyn. S., Vol. 32, pp. 3099-3131, 2012

    work page 2012

  16. [16]

    Lyu, P.J

    J. Lyu, P.J. Schofield, K. M. Reaver, M.A. Beauregard, R.D. Parshad, A comparison of the Trojan Y Chromosome strategy to harvesting models for eradication of non-native species , arXiv:1810.08279

    work page arXiv

  17. [17]

    Quittner, P., & Souplet, P. (2007). Superlinear parabolic problems. Blow-up, Global Existence and Steady States, Birkhuser Advanced Texts, Birkhuser, Basel

    work page 2007

  18. [18]

    Myers, D

    J.H. Myers, D. Simberloff, A.M. Kuris and J.R. Carey, Eradication revisited: dealing with exotic species, Trends Ecol. Evol., Vol. 15, pp. 316-320, 2000

    work page 2000

  19. [19]

    Okubo, P.K

    A. Okubo, P.K. Maini, M.H. Williamson and J.D. Murray, The spread of the grey squirrel in Britain, P. R. Soc. Lond. B Bio, Vol. 238, pp. 113-125, 1989

    work page 1989

  20. [20]

    R. D. Parshad, On the long time behavior of a PDE model for invasive species control , Int. J. Mathemat. Anal., Vol. 5, No. 40, pp. 1991-2015, 2011

    work page 1991

  21. [21]

    R. D. Parshad and J. B. Gutierrez, On the Global Attractor of the Trojan Y-Chromosome Model, Commun. Pur. Appl. Anal., Vol. 10, pp. 339-359, 2011

    work page 2011

  22. [22]

    R. D. Parshad and J. B. Gutierrez, On the well posedness of the Trojan Y Chromosome model, Bound. Value Probl., Vol. 2010, Article ID 405816, pp. 1-29, 2010

    work page 2010

  23. [23]

    R. D. Parshad, S. Kouachi and J. B. Gutierrez, Global existence and asymptotic behavior of a model for biological control of invasive species via supermale introduction , Communications in Mathematical Sciences, Vol. 11, No. 4, pp. 951-972, 2013

    work page 2013

  24. [24]

    Perrin, Sex reversal: a fountain of youth for sex chromosomes ? Evolution, vol 63, pp

    N. Perrin, Sex reversal: a fountain of youth for sex chromosomes ? Evolution, vol 63, pp. 3043-3049, 2009

    work page 2009

  25. [25]

    D. J. Schill, K.A. Meyer, and M. J. Hansen,Simulated effects of YY-male stocking and manual suppression for eradicating nonnative brook trout populations , N. Am. J. Fish. Manage., 37:10541066, 2017. 18 TYC BLOW-UP

    work page 2017

  26. [26]

    Schill, J.A

    D.J. Schill, J.A. Heindel, M.R. Campbell, K.A. Meyer & E.R. Mamer, Production of a YY Male Brook Trout Broodstock for Potential Eradication of Undesired Brook Trout Popula- tions. N. Am. J. Aquacult., 78(1), pp. 72-83, 2016

    work page 2016

  27. [27]

    Schofield and W

    P. Schofield and W. Loftus, Nonnative fishes in Florida freshwaters: a literature review and synthesis, Rev. Fish Biol. Fisher., Vol. 25, No. 1, pp. 117-145

    work page

  28. [28]

    93-106, 2000

    Pierre, M., & Schmitt, D., Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev, 42(1), pp. 93-106, 2000

    work page 2000

  29. [29]

    Math., 78(2), pp

    Pierre, M., Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78(2), pp. 417-455, 2010

    work page 2010

  30. [30]

    Biological invasions:Theory and practice

    N. Shigesada and K. Kawasaki, “Biological invasions:Theory and practice”, Oxford University Press, Oxford, 1997

    work page 1997

  31. [31]

    Teem, J.B

    J.L. Teem, J.B. Gutierrez and R.D. Parshad, A comparison of the Trojan Y Chromosome model and daughterless carp eradication strategies , Bio. Invasions, Vol. 16, No. 6, pp. 1217- 1230, 2014

    work page 2014

  32. [32]

    Tianran, W

    Z. Tianran, W. Wang, M athematical models of two-sex population dynamics, Kˆ okyˆuroku, Vol. 1432, pp. 96-104, 2005

    work page 2005

  33. [33]

    Biological Control

    R. Van Driesche and T. Bellows “Biological Control”, Kluwer Academic Publishers, Mas- sachusetts, 1996

    work page 1996

  34. [34]

    X. Wang, R. D. Parshad and J. Walton, The Stochastic Trojan Y Chromosome model for eradication of an invasive species , J. Biol. Dynam., Vol. 10, Issue 1, pp. 179-199, 2016

    work page 2016

  35. [35]

    B., Hurdal, M

    Gutierrez, J. B., Hurdal, M. K., Parshad, R. D., & Teem, J. L., Analysis of the Trojan Y chromosome model for eradication of invasive species in a dendritic riverine system, J. Math. Biol., 64(1-2), pp. 319-340, 2012

    work page 2012

  36. [36]

    X. Wang, J. R. Walton, R. D. Parshad, K. Storey and M. Boggess, Analysis of Trojan Y- Chromosome eradication strategy, J. Math. Biol., 68(7), pp. 1731-1756, 2014

    work page 2014

  37. [37]

    X. Zhao, B. Liu and D. Ning, Existence of global attractor for the Trojan Y Chromosome model, Electron. J. Qual. Theo., No. 36, 16, 2012

    work page 2012