doi: 10.3934/dcdsb.2021260
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Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Guohong Zhang

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: G.-H. Zhang is partially supported by grants from National Science Foundation of China(11871403); X.-L. Wang is partially supported by grants from Fundamental Research Funds for the Central Universities (XDJK2020B050)

In this paper, we investigate the global dynamics of a Leslie-Gower predator-prey system in advective homogeneous environments. We discuss the existence and uniqueness of positive steady-state solutions. We study the large time behavior of solutions and establish threshold conditions for persistence and extinction of two species when they live in open advective environments. Numerical simulations indicate that the introduction of advection leads to the evolution of spatial distribution patterns of species and specially it may induce spatial separation of the prey and predator under some conditions.

Citation: Baifeng Zhang, Guohong Zhang, Xiaoli Wang. Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021260
References:
[1]

T. Agrawal and M. Saleem, Complex dynamics in a ratio-dependent two-predator one-prey model, Comput. Appl. Math., 34 (2015), 265-274.  doi: 10.1007/s40314-014-0115-1.  Google Scholar

[2]

M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model, Chaos Solitons Fractals, 14 (2002), 1275-1293.  doi: 10.1016/S0960-0779(02)00079-6.  Google Scholar

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M. A. Aziz-Alaoui and M. D. 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

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M. BallykL. DungD. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.  Google Scholar

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A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.  Google Scholar

[6]

Y. CaiC. ZhaoW. Wang and J. Wang, Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Model., 39 (2015), 2092-2106.  doi: 10.1016/j.apm.2014.09.038.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

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F. ChenL. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. RWA, 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009.  Google Scholar

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S. Chen and J. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.  doi: 10.1016/j.aml.2011.09.070.  Google Scholar

[10]

S. Chen, J. Shi and J. Wei, Global stability and hopf bifurcation in a delayed diffusion Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos, 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

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K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.  Google Scholar

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M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Popul. Biol., 67 (2005), 33-45.  doi: 10.1016/j.tpb.2004.07.005.  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

[14]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

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Y. Du and M. Wang, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 759-778.  doi: 10.1017/S0308210500004704.  Google Scholar

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J. Giné and C. Valls, Nonlinear oscillations in the modified Leslie-Gower model, Nonlinear Anal. RWA, 51 (2020), 103010.  doi: 10.1016/j.nonrwa.2019.103010.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Berlin, 2008.  Google Scholar

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F. M. Hilker and M. A. Lewis, Predator-prey systems in streams and rivers, Theor. Ecol., 3 (2010), 175-193.  doi: 10.1007/s12080-009-0062-4.  Google Scholar

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C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[21]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[22]

S.-B. Hsu and T.-W. Huang, Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type, Taiwanese J. Math., 3 (1999), 35-53.  doi: 10.11650/twjm/1500407053.  Google Scholar

[23]

M. G. Kre$\mathop {\rm{i}}\limits^ \vee $n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Usp. Mat. Nauk., 3 (1948), 3-95.   Google Scholar

[24]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

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P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.  doi: 10.1093/biomet/35.3-4.213.  Google Scholar

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P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.  doi: 10.1093/biomet/45.1-2.16.  Google Scholar

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C. LetellierL. A. AguirreJ. Maquet and M. A. Aziz-Alaoui, Should all the species of a food chain be counted to investigate the global dynamics?, Chaos Solitons Fractals, 13 (2002), 1099-1113.  doi: 10.1016/S0960-0779(01)00116-3.  Google Scholar

[28]

C. Letellier and M. A. Aziz-Alaoui, Analysis of the dynamics of a realistic ecological model, Chaos Solitons Fractals, 13 (2002), 95-107.  doi: 10.1016/S0960-0779(00)00239-3.  Google Scholar

[29]

X. LiW. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA. J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.  Google Scholar

[30]

Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34 (2007), 606-620.  doi: 10.1016/j.chaos.2006.03.068.  Google Scholar

[31]

Z. Liang and H. Pan, Qualitative analysis of a ratio-dependent holling-tanner model, J. Math. Anal. Appl., 334 (2007), 954-964.  doi: 10.1016/j.jmaa.2006.12.079.  Google Scholar

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J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonlinear Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.  Google Scholar

[33]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, 1925. Google Scholar

[34]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar

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Y. LouH. Nie and Y. Wang, Coexistence and bistability of a competition model in open advective environments, Math. Biosci., 306 (2018), 10-19.  doi: 10.1016/j.mbs.2018.09.013.  Google Scholar

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Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar

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Z.-P. Ma, Spatiotemporal dynamics of a diffusive Leslie-Gower prey-predator model with strong Allee effect, Nonlinear Anal. RWA, 50 (2019), 651-674.  doi: 10.1016/j.nonrwa.2019.06.008.  Google Scholar

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W. Ni and M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 7 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[39]

H. NieB. Wang and J. Wu, Invasion analysis on a predator-prey system in open advective environments, J. Math. Biol., 81 (2020), 1429-1463.  doi: 10.1007/s00285-020-01545-3.  Google Scholar

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A. F. Nindjin and M. A. Aziz-Alaoui, Persistence and global stability in a delayed Leslie-Gower type three species food chain, J. Math. Anal. Appl., 340 (2008), 340-357.  doi: 10.1016/j.jmaa.2007.07.078.  Google Scholar

[41]

A. F. NindjinM. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal. RWA, 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003.  Google Scholar

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L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[43]

R. Peng and M. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Appl. Math. Lett., 20 (2007), 664-670.  doi: 10.1016/j.aml.2006.08.020.  Google Scholar

[44]

R. Peng and M. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.  doi: 10.1017/S0308210500003814.  Google Scholar

[45]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.  Google Scholar

[46]

O. Vasilyeva and F. Lutscher, How flow speed alters competitive outcome in advective environments, Bull. Math. Biol., 74 (2012), 2935-2958.  doi: 10.1007/s11538-012-9792-3.  Google Scholar

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V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

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M. WangP. Y. H. Pang and W. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA. J. Appl. Math., 73 (2008), 815-835.  doi: 10.1093/imamat/hxn016.  Google Scholar

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N. Zhang, F. Chen, Q. Su and T. Wu, Dynamic behaviors of a harvesting Leslie-Gower predator-prey model, Discrete Dyn. Nat. Soc., (2011), 473949. doi: 10.1155/2011/473949.  Google Scholar

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X.-Q. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differ. Equ., 55 (2016), 73.  doi: 10.1007/s00526-016-1021-8.  Google Scholar

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J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, J. Math. Anal. Appl., 389 (2012), 1380-1393.  doi: 10.1016/j.jmaa.2012.01.013.  Google Scholar

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show all references

References:
[1]

T. Agrawal and M. Saleem, Complex dynamics in a ratio-dependent two-predator one-prey model, Comput. Appl. Math., 34 (2015), 265-274.  doi: 10.1007/s40314-014-0115-1.  Google Scholar

[2]

M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model, Chaos Solitons Fractals, 14 (2002), 1275-1293.  doi: 10.1016/S0960-0779(02)00079-6.  Google Scholar

[3]

M. A. Aziz-Alaoui and M. D. 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]

M. BallykL. DungD. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.  Google Scholar

[5]

A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.  Google Scholar

[6]

Y. CaiC. ZhaoW. Wang and J. Wang, Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Model., 39 (2015), 2092-2106.  doi: 10.1016/j.apm.2014.09.038.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

F. ChenL. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. RWA, 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009.  Google Scholar

[9]

S. Chen and J. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.  doi: 10.1016/j.aml.2011.09.070.  Google Scholar

[10]

S. Chen, J. Shi and J. Wei, Global stability and hopf bifurcation in a delayed diffusion Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos, 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

[11]

K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.  Google Scholar

[12]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Popul. Biol., 67 (2005), 33-45.  doi: 10.1016/j.tpb.2004.07.005.  Google Scholar

[13]

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

[14]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

[15]

Y. Du and M. Wang, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 759-778.  doi: 10.1017/S0308210500004704.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[17]

J. Giné and C. Valls, Nonlinear oscillations in the modified Leslie-Gower model, Nonlinear Anal. RWA, 51 (2020), 103010.  doi: 10.1016/j.nonrwa.2019.103010.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Berlin, 2008.  Google Scholar

[19]

F. M. Hilker and M. A. Lewis, Predator-prey systems in streams and rivers, Theor. Ecol., 3 (2010), 175-193.  doi: 10.1007/s12080-009-0062-4.  Google Scholar

[20]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[21]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[22]

S.-B. Hsu and T.-W. Huang, Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type, Taiwanese J. Math., 3 (1999), 35-53.  doi: 10.11650/twjm/1500407053.  Google Scholar

[23]

M. G. Kre$\mathop {\rm{i}}\limits^ \vee $n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Usp. Mat. Nauk., 3 (1948), 3-95.   Google Scholar

[24]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

[25]

P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.  doi: 10.1093/biomet/35.3-4.213.  Google Scholar

[26]

P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.  doi: 10.1093/biomet/45.1-2.16.  Google Scholar

[27]

C. LetellierL. A. AguirreJ. Maquet and M. A. Aziz-Alaoui, Should all the species of a food chain be counted to investigate the global dynamics?, Chaos Solitons Fractals, 13 (2002), 1099-1113.  doi: 10.1016/S0960-0779(01)00116-3.  Google Scholar

[28]

C. Letellier and M. A. Aziz-Alaoui, Analysis of the dynamics of a realistic ecological model, Chaos Solitons Fractals, 13 (2002), 95-107.  doi: 10.1016/S0960-0779(00)00239-3.  Google Scholar

[29]

X. LiW. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA. J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.  Google Scholar

[30]

Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34 (2007), 606-620.  doi: 10.1016/j.chaos.2006.03.068.  Google Scholar

[31]

Z. Liang and H. Pan, Qualitative analysis of a ratio-dependent holling-tanner model, J. Math. Anal. Appl., 334 (2007), 954-964.  doi: 10.1016/j.jmaa.2006.12.079.  Google Scholar

[32]

J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonlinear Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.  Google Scholar

[33]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, 1925. Google Scholar

[34]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar

[35]

Y. LouH. Nie and Y. Wang, Coexistence and bistability of a competition model in open advective environments, Math. Biosci., 306 (2018), 10-19.  doi: 10.1016/j.mbs.2018.09.013.  Google Scholar

[36]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar

[37]

Z.-P. Ma, Spatiotemporal dynamics of a diffusive Leslie-Gower prey-predator model with strong Allee effect, Nonlinear Anal. RWA, 50 (2019), 651-674.  doi: 10.1016/j.nonrwa.2019.06.008.  Google Scholar

[38]

W. Ni and M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 7 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[39]

H. NieB. Wang and J. Wu, Invasion analysis on a predator-prey system in open advective environments, J. Math. Biol., 81 (2020), 1429-1463.  doi: 10.1007/s00285-020-01545-3.  Google Scholar

[40]

A. F. Nindjin and M. A. Aziz-Alaoui, Persistence and global stability in a delayed Leslie-Gower type three species food chain, J. Math. Anal. Appl., 340 (2008), 340-357.  doi: 10.1016/j.jmaa.2007.07.078.  Google Scholar

[41]

A. F. NindjinM. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal. RWA, 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003.  Google Scholar

[42]

L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar

[43]

R. Peng and M. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Appl. Math. Lett., 20 (2007), 664-670.  doi: 10.1016/j.aml.2006.08.020.  Google Scholar

[44]

R. Peng and M. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.  doi: 10.1017/S0308210500003814.  Google Scholar

[45]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.  Google Scholar

[46]

O. Vasilyeva and F. Lutscher, How flow speed alters competitive outcome in advective environments, Bull. Math. Biol., 74 (2012), 2935-2958.  doi: 10.1007/s11538-012-9792-3.  Google Scholar

[47]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

[48]

M. WangP. Y. H. Pang and W. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA. J. Appl. Math., 73 (2008), 815-835.  doi: 10.1093/imamat/hxn016.  Google Scholar

[49]

Y. WangJ. Shi and J. Wang, Persistence and extinction of population in reaction-diffusion-advection model with strong Allee effect growth, J. Math. Biol., 78 (2019), 2093-2140.  doi: 10.1007/s00285-019-01334-7.  Google Scholar

[50]

N. Zhang, F. Chen, Q. Su and T. Wu, Dynamic behaviors of a harvesting Leslie-Gower predator-prey model, Discrete Dyn. Nat. Soc., (2011), 473949. doi: 10.1155/2011/473949.  Google Scholar

[51]

X.-Q. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differ. Equ., 55 (2016), 73.  doi: 10.1007/s00526-016-1021-8.  Google Scholar

[52]

J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, J. Math. Anal. Appl., 389 (2012), 1380-1393.  doi: 10.1016/j.jmaa.2012.01.013.  Google Scholar

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Figure 1.  Bifurcation diagrams of steady state solutions to (3) in open advective environments with the bifurcation parameter $ q $. The fixed parameter values $ b = 1 $, $ a = 1, c = 10, L = 10 $. (a) $ d_1 = d_2 = 0.01,r_1 = 5,r_2 = 6 $; (b) $ d_1 = 0.1,d_2 = 0.15,r_1 = r_2 = 5 $; (c) $ d_1 = d_2 = 0.1,r_1 = r_2 = 16 $; (d) $ d_1 = d_2 = 0.15,r_1 = 5,r_2 = 2 $; (e) $ d_1 = 0.15,d_2 = 0.1,r_1 = r_2 = 12 $
Figure 2.  Evolution of spatial distribution patterns on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 1 $ and $ q^*(d_1,r_1,b)\le q^*(d_2,r_2,b) $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.1,d_2 = 0.15,r_1 = r_2 = 5 $.(a) $ q = 1.7 $, (b) $ q = 1.9 $, (c) $ q = 2 $
Figure 3.  Evolution of spatial distribution patterns on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 1 $ and $ q^*(d_1,r_1,b)>q^*(d_2,r_2,b) $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.15,d_2 = 0.1,r_1 = r_2 = 12 $. (a) $ q = 1.5 $, (b) $ q = 3.3 $, (c) $ q = 3.38 $, (d) $ q = 3.8 $, (e)$ q = 3.875 $, (f) $ q = 4 $
Figure 4.  Bifurcation diagrams of steady state solutions to (3) with the bifurcation parameter $ q $ when the advective environments is closed. The fixed parameter values $ b = 0, $ $ a = 1, c = 10, L = 10 $, $ d_1 = 0.15,d_2 = 0.1,r_1 = 30,r_2 = 9 $
Figure 5.  Evolution of the stable steady-state spatial distribution patterns of species on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 0 $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.15,d_2 = 0.1,r_1 = 30,r_2 = 9 $. (a) $ q = 0.1 $, (b) $ q = 2.5 $, (c) $ q = 3 $, (d) $ q = 7.2 $, (e) $ q = 20 $
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