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doi: 10.3934/dcdsb.2021129
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A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

2. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author: Hongmei Cheng

Received  January 2021 Early access April 2021

Fund Project: The second author is supported by NSFC grant 11701341. The third author is supported by NSFC grant 11771044

This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region $ [0,h_0] $. We first show that the problem has a unique solution defined for all $ t>0 $. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria.

Citation: Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021129
References:
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D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 466 (1975), 5-49.   Google Scholar

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D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[4]

P. W. Bates and G. Zhou, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[5]

H. BerestyckiJ. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.  Google Scholar

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G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media., 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

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J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  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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H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579.  doi: 10.1515/fca-2015-0035.  Google Scholar

[10]

H. Cheng and R. Yuan, Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response, Taiwanese J. Math., 20 (2016), 801-822.  doi: 10.11650/tjm.20.2016.6284.  Google Scholar

[11]

H. Cheng and R. Yuan, Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3007-3022.  doi: 10.3934/dcdsb.2017160.  Google Scholar

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H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 5433-5454.  doi: 10.3934/dcds.2017236.  Google Scholar

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H. Cheng and R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion, Appl. Math. Comput., 338 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.049.  Google Scholar

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H. Cheng and S. Niu, A free boundary problem of some Lesile-Gower predator-prey model with higher dimensional environment, submitted, (2019). Google Scholar

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W. DingR. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.  Google Scholar

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Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

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Y. DuZ. 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

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Y. 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

[20]

Y. Du and Z. 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

[21]

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

[22]

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

[23]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.  Google Scholar

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

[25]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[26]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.  Google Scholar

[27]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar

[28]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[29]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[30]

Y. LiuZ GuoM. El Smaily and L. Wang, A Leslie-Gower predator-prey model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2063-2084.  doi: 10.3934/dcdss.2019133.  Google Scholar

[31]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.  Google Scholar

[32]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[33]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17(1987), 241–280. doi: 10.32917/hmj/1206130066.  Google Scholar

[34]

S. Niu and H. Cheng, A free boundary problem for a Leslie-Gower predator-prey model in higher dimensions and heterogeneous environment, American Journal of Applied Mathematics, 8 (2020), 284–292. Google Scholar

[35]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wave fronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[36]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.  Google Scholar

[37]

N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8.  Google Scholar

[38]

N. Sun, A time-periodic reaction-diffusion-advection equation with a free boundary and sign-changing coefficients, Nonlinear Anal. Real World Appl., 51 (2020), 102952, 28 pp. doi: 10.1016/j.nonrwa.2019.06.002.  Google Scholar

[39]

N. Sun and X. Han, Asymptotic behavior of solutions of a reaction-diffusion model with a protection zone and a free boundary, Appl. Math. Lett., 107 (2020), 106470, 7 pp. doi: 10.1016/j.aml.2020.106470.  Google Scholar

[40]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch. Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988.  Google Scholar

[41]

M. 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

[42]

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

[43]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[44]

B. Yan and C. An, The sign-changing solutions for a class of nonlocal elliptic problem in an annulus, Topol. Methods Nonlinear Anal, 55 (2020), 1-18.  doi: 10.12775/tmna.2019.081.  Google Scholar

[45]

R. Yang and J. Wei, The effect of delay on a diffusive predator-prey system with modified Leslie-Gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7.  Google Scholar

[46]

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

[47]

J. Zhao and M. 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

[48]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

[49]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.  Google Scholar

[50]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 466 (1975), 5-49.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[4]

P. W. Bates and G. Zhou, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[5]

H. BerestyckiJ. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.  Google Scholar

[6]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media., 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[7]

J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

[8]

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

[9]

H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579.  doi: 10.1515/fca-2015-0035.  Google Scholar

[10]

H. Cheng and R. Yuan, Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response, Taiwanese J. Math., 20 (2016), 801-822.  doi: 10.11650/tjm.20.2016.6284.  Google Scholar

[11]

H. Cheng and R. Yuan, Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3007-3022.  doi: 10.3934/dcdsb.2017160.  Google Scholar

[12]

H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 5433-5454.  doi: 10.3934/dcds.2017236.  Google Scholar

[13]

H. Cheng and R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion, Appl. Math. Comput., 338 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.049.  Google Scholar

[14]

H. Cheng and S. Niu, A free boundary problem of some Lesile-Gower predator-prey model with higher dimensional environment, submitted, (2019). Google Scholar

[15]

W. DingR. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.  Google Scholar

[16]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[17]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar

[18]

Y. DuZ. 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

[19]

Y. 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

[20]

Y. Du and Z. 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

[21]

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

[22]

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

[23]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.  Google Scholar

[24]

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

[25]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[26]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.  Google Scholar

[27]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar

[28]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[29]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[30]

Y. LiuZ GuoM. El Smaily and L. Wang, A Leslie-Gower predator-prey model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2063-2084.  doi: 10.3934/dcdss.2019133.  Google Scholar

[31]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.  Google Scholar

[32]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[33]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17(1987), 241–280. doi: 10.32917/hmj/1206130066.  Google Scholar

[34]

S. Niu and H. Cheng, A free boundary problem for a Leslie-Gower predator-prey model in higher dimensions and heterogeneous environment, American Journal of Applied Mathematics, 8 (2020), 284–292. Google Scholar

[35]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wave fronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[36]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.  Google Scholar

[37]

N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8.  Google Scholar

[38]

N. Sun, A time-periodic reaction-diffusion-advection equation with a free boundary and sign-changing coefficients, Nonlinear Anal. Real World Appl., 51 (2020), 102952, 28 pp. doi: 10.1016/j.nonrwa.2019.06.002.  Google Scholar

[39]

N. Sun and X. Han, Asymptotic behavior of solutions of a reaction-diffusion model with a protection zone and a free boundary, Appl. Math. Lett., 107 (2020), 106470, 7 pp. doi: 10.1016/j.aml.2020.106470.  Google Scholar

[40]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch. Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988.  Google Scholar

[41]

M. 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

[42]

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

[43]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[44]

B. Yan and C. An, The sign-changing solutions for a class of nonlocal elliptic problem in an annulus, Topol. Methods Nonlinear Anal, 55 (2020), 1-18.  doi: 10.12775/tmna.2019.081.  Google Scholar

[45]

R. Yang and J. Wei, The effect of delay on a diffusive predator-prey system with modified Leslie-Gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7.  Google Scholar

[46]

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

[47]

J. Zhao and M. 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

[48]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

[49]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.  Google Scholar

[50]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.  Google Scholar

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