Skip to main content
SpringerLink
Log in

Invasive behaviour under competition via a free boundary model: a numerical approach

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

What will happen when two invasive species are competing and invading the environment at the same time? In this paper, we try to find all the possible scenarios in such a situation based on the diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a “chase-and-run coexistence” phenomenon, for certain parameter ranges when the initial functions are chosen properly. Here we use a numerical approach to extend the theoretical research of Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) in several directions. Firstly, we examine how the longtime dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible longtime profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species invade successfully. Secondly, we show through numerical experiments that the basic features of the model appear to be retained when the environment is perturbed by periodic variation in time. Thirdly, our numerical analysis suggests that in two space dimensions the population range and the spatial population distribution of the successful invader tend to become more and more circular as time increases no matter what geometrical shape the initial population range possesses. Our numerical simulations cover the one space dimension case, and two space dimension case with or without spherical symmetry. The numerical methods here are based on that of Liu et al. (Mathematics, 6(5):72, 2018, Int J Comput Math, 97(5): 959–979, 2020). In the two space dimension case without radial symmetry, the level set method is used, while the front tracking method is used for the remaining cases. We hope the numerical observations in this paper can provide further insights to the biological invasion problem, and also to future theoretical investigations. More importantly, we hope the numerical analysis may reach more biologically oriented experts and inspire applications of some refined versions of the model tailored to specific real world biological invasion problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54
Fig. 55
Fig. 56

Similar content being viewed by others

Notes

  1. Which means \(s_2(t)\equiv \infty \) and \(\inf _{r\ge 0}v(r,0)>0\). See Du et al. (2017) for precise statement.

  2. Here we have used (1.11) to estimate \( (c^*_{\mu _1}, s^*_{\mu _2}) \) by \( (s_1'(t), s_2'(t))\) for large enough t, with a suitably chosen initial function pair.

References

  • Bunting G, Du Y, Krakowski K (2012) Spreading speed revisited: analysis of a free boundary model. Netw Heterog Med 7(4):583–603

    Article  MathSciNet  Google Scholar 

  • de Mottoni P (1979) Qualitative analysis for some quasilinear parabolic systems. Inst Math Pol Acad Sci Zam 190:11–79

    Google Scholar 

  • Du Y, Guo Z (2011) Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary. II. J Differ Equ 250(12):4336–4366

    Article  MathSciNet  Google Scholar 

  • Du Y, Guo Z (2012) The Stefan problem for the Fisher-KPP equation. J Differ Equ 253:996–1035

    Article  MathSciNet  Google Scholar 

  • Du Y, Lin Z (2010) Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal 42(1):377–405

    Article  MathSciNet  Google Scholar 

  • Du Y, Lin Z (2014) The diffusive competition model with a free boundary: invasion of a superior or inferior competitor. Discrete Contin Dyn Syst B 19(10):3105–3132

    Article  MathSciNet  Google Scholar 

  • Du Y, Lou B (2015) Spreading and vanishing in nonlinear diffusion problems with free boundaries. J Eur Math Soc 17(10):2673–2724

    Article  MathSciNet  Google Scholar 

  • Du Y, Matano H, Wang K (2014) Regularity and asymptotic behavior of nonlinear Stefan problems. Arch Rational Mech Anal 212:957–1010

    Article  MathSciNet  Google Scholar 

  • Du Y, Matsuzawa H, Zhou M (2015) Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIAM J Math Anal 46:375–396

    Article  MathSciNet  Google Scholar 

  • Du Y, Matsuzawa H, Zhou M (2015) Spreading speed and profile for nonlinear Stefan problems in high space dimensions. J Math Pures Appl 103:741–787

    Article  MathSciNet  Google Scholar 

  • Du Y, Wang M, Zhou M (2017) Semi-wave and spreading speed for the diffusive competition model with a free boundary. J Math Pures Appl 107(3):253–287

    Article  MathSciNet  Google Scholar 

  • Du Y, Wu C-H (2018) Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries. Calc Var Partial Differ Equ 57(2):52

    Article  MathSciNet  Google Scholar 

  • Fisher FA (1937) The wave of advance of advantageous genes. Ann Eugen 7:335–369

    MATH  Google Scholar 

  • Girardin L, Lam KY (2019) Invasion of open space by two competitors: spreading properties of monostable two-species competition-diffusion systems. Proc Lond Math Soc 119:1279–1335

    Article  MathSciNet  Google Scholar 

  • Guo J-S, Wu C-H (2012) On a free boundary problem for a two-species weak competition system. J Dyn Differ Equ 24(4):873–895

    Article  MathSciNet  Google Scholar 

  • Guo J-S, Wu C-H (2015) Dynamics for a two-species competition-diffusion model with two free boundaries. Nonlinearity 28(1):1–27

    Article  MathSciNet  Google Scholar 

  • Kearney M, Phillips BL, Tracy CR, Christian KA, Betts G, Porter WP (2008) Modelling species distributions without using species distributions: the cane toad in Australia under current and future climates. Ecography 31(4):423–434

    Article  Google Scholar 

  • Khan K, Liu S, Schaerf T, Du Y (2020) Invasive behaviour under competition based on a model with diffusion and free boundaries: A numerical approach, preprint (http://turing.une.edu.au/~ydu/papers/Comp-numerical.pdf)

  • Kolmogorov AN, Petrovski IG, Piskunov NS (1937) A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull Mosc Univ Math Mech 1:1–25

    Google Scholar 

  • Liu S, Du Y, Liu X (2020) Numerical studies of a class of reaction-diffusion equations with Stefan conditions. Int J Comput Math 97(5):959–979

    Article  MathSciNet  Google Scholar 

  • Liu S, Liu X (2018) Numerical methods for a two-species competition-diffusion model with free boundaries. Mathematics 6(5):72

    Article  MathSciNet  Google Scholar 

  • Liu S, Liu X (2020) Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Disc Contin Dyn Syst B 25(1):141–159

    MathSciNet  MATH  Google Scholar 

  • Okubo A, Maini PK, Williamson MH, Murray JD (1989) On the spatial spread of the grey squirrel in Britain. Proc R Soc Lond B 238(1291):113–125

    Article  Google Scholar 

  • Piqueras M-A, Company R, Jódar L (2017) A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model. J Comput Appl Math 309:473–481

    Article  MathSciNet  Google Scholar 

  • Shine R (2014) A review of ecological interactions between native frogs and invasive cane toads in Australia. Austral Ecol 39(1):1–16

    Article  Google Scholar 

  • Tian C, Ruan S (2018) On an advection-reaction-diffusion competition system with double free boundaries modeling invasion and competition of Aedes albopictus and Aedes aegypti mosquitoes. J Differ Equ 265:4016–4051

    Article  MathSciNet  Google Scholar 

  • Wang MX, Zhang Y (2017) Note on a two-species competition-diffusion model with two free boundaries. Nonl Anal 159:458–467

    Article  MathSciNet  Google Scholar 

  • Wang Z, Nie H, Du Y (2019) Asymptotic spreading speed for the weak competition system with a free boundary. Discrete Contin Dyn Syst Ser A 39:5223–5262

    Article  MathSciNet  Google Scholar 

  • Wu C-H (2013) Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete Contin Dyn Syst Ser B 18(9):2441–2455

    MathSciNet  MATH  Google Scholar 

  • Wu C-H (2015) The minimal habitat size for spreading in a weak competition system with two free boundaries. J Differ Equ 259(3):873–897

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia

    Kamruzzaman Khan, Timothy M. Schaerf & Yihong Du

  2. Department of Mathematics, University of South Carolina, Columbia, USA

    Shuang Liu

  3. Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

    Kamruzzaman Khan

Authors
  1. Kamruzzaman Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Timothy M. Schaerf
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yihong Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yihong Du.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, K., Liu, S., Schaerf, T.M. et al. Invasive behaviour under competition via a free boundary model: a numerical approach. J. Math. Biol. 83, 23 (2021). https://doi.org/10.1007/s00285-021-01641-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00285-021-01641-y

Keywords

Mathematics Subject Classification

Search

Navigation