May  2018, 38(5): 2591-2607. doi: 10.3934/dcds.2018109

Dynamics for the diffusive Leslie-Gower model with double free boundaries

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang

Received  July 2017 Revised  January 2018 Published  March 2018

Fund Project: This work was supported by NSFC Grants 11771110 and 11371113.

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

Citation: Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
References:
[1]

H. Bunting, Y. H. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media (special issue dedicated to H. Matano), 7(2012), 583-603. doi: 10.3934/nhm.2012.7.583.  Google Scholar

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J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.  Google Scholar

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S. S. Chen, J. P. Shi and J. J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

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Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

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Y. H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.  Google Scholar

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Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

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Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc. (JEMS), 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

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Y. H. DuM. X. Wang and M. L. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

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H. GuB. D. Lou and M. L. Zhou, Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar

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J. S. Guo and C.-H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

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[16]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

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P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.  Google Scholar

[19]

H. Monobe and C.-H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differential Equations, 261 (2016), 6144-6177.  doi: 10.1016/j.jde.2016.08.033.  Google Scholar

[20]

Y. W. Qi and Y. Zhu, Global stability of Lesile-type predator-prey model, Methods and Applications of Analysis, 23 (2016), 259-268.  doi: 10.4310/MAA.2016.v23.n3.a3.  Google Scholar

[21]

N. K. Sun, B. D. Lou and M. L. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Diff. Equ., 56 (2017), Art. 61, 36 pp. doi: 10.1007/s00526-017-1165-1..  Google Scholar

[22]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[23]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.  Google Scholar

[24]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[25]

M. X. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.  Google Scholar

[26]

M. WangW. J. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329.  doi: 10.1016/j.jmaa.2016.04.007.  Google Scholar

[27]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.: Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[28]

M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[29]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal: TMA, 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[30]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differental Equatons, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[31]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[32]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[33]

M. X. Wang and Y. G. Zhao, A semilinear parabolic system with a free boundary, Z. Angew. Math. Phys., 66 (2015), 3309-3332.  doi: 10.1007/s00033-015-0582-2.  Google Scholar

[34]

L. Wei, G. H. Zhang and M. L. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Diff. Equ., 55 (2016), Art. 95, 34 pp. doi: 10.1007/s00526-016-1039-y.  Google Scholar

[35]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[36]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Applicable Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.  Google Scholar

[37]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal.: Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[38]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.  Google Scholar

[39]

Y. G. Zhao and M. X. Wang, Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters, Nonlinear Anal.: Real World Appl., 31 (2016), 166-178.  doi: 10.1016/j.nonrwa.2016.02.001.  Google Scholar

[40]

L. ZhouS. Zhang and Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. Royal Soc. Edinburgh A, 147 (2017), 615-648.  doi: 10.1017/S0308210516000226.  Google Scholar

show all references

References:
[1]

H. Bunting, Y. H. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media (special issue dedicated to H. Matano), 7(2012), 583-603. doi: 10.3934/nhm.2012.7.583.  Google Scholar

[2]

J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.  Google Scholar

[3]

S. S. Chen, J. P. Shi and J. J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

[4]

Y. H. DuZ. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar

[5]

Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[6]

Y. H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.  Google Scholar

[7]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[8]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst.-B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[9]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc. (JEMS), 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

[10]

Y. H. DuM. X. Wang and M. L. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[11]

J. GeI. K. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar

[12]

H. GuB. D. Lou and M. L. Zhou, Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar

[13]

J. S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[14]

J. S. Guo and C.-H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[15]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for reaction-advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.  Google Scholar

[16]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

[17] O. A. LadyzenskajaU. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York, London, 1968.   Google Scholar
[18]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.  Google Scholar

[19]

H. Monobe and C.-H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differential Equations, 261 (2016), 6144-6177.  doi: 10.1016/j.jde.2016.08.033.  Google Scholar

[20]

Y. W. Qi and Y. Zhu, Global stability of Lesile-type predator-prey model, Methods and Applications of Analysis, 23 (2016), 259-268.  doi: 10.4310/MAA.2016.v23.n3.a3.  Google Scholar

[21]

N. K. Sun, B. D. Lou and M. L. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Diff. Equ., 56 (2017), Art. 61, 36 pp. doi: 10.1007/s00526-017-1165-1..  Google Scholar

[22]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[23]

M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.  Google Scholar

[24]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[25]

M. X. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.  Google Scholar

[26]

M. WangW. J. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329.  doi: 10.1016/j.jmaa.2016.04.007.  Google Scholar

[27]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.: Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[28]

M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[29]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal: TMA, 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[30]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differental Equatons, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[31]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[32]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[33]

M. X. Wang and Y. G. Zhao, A semilinear parabolic system with a free boundary, Z. Angew. Math. Phys., 66 (2015), 3309-3332.  doi: 10.1007/s00033-015-0582-2.  Google Scholar

[34]

L. Wei, G. H. Zhang and M. L. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Diff. Equ., 55 (2016), Art. 95, 34 pp. doi: 10.1007/s00526-016-1039-y.  Google Scholar

[35]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[36]

Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Applicable Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.  Google Scholar

[37]

J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal.: Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[38]

Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.  Google Scholar

[39]

Y. G. Zhao and M. X. Wang, Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters, Nonlinear Anal.: Real World Appl., 31 (2016), 166-178.  doi: 10.1016/j.nonrwa.2016.02.001.  Google Scholar

[40]

L. ZhouS. Zhang and Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. Royal Soc. Edinburgh A, 147 (2017), 615-648.  doi: 10.1017/S0308210516000226.  Google Scholar

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