November  2020, 25(11): 4189-4210. doi: 10.3934/dcdsb.2020093

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

1. 

School of Information and statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

2. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Rong Zou

Received  June 2019 Revised  November 2019 Published  April 2020

Fund Project: The second author is supported by NSF of China (Grants No. 11671123).

In this paper, we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment. The global existence and boundedness of solutions are shown. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global stability of semi-trivial solutions. The existence of positive steady state solution bifurcating from semi-trivial solutions is obtained by using local bifurcation theory. The stability analysis of the positive steady state solution is investigated in detail. In addition, we explore the asymptotic profiles of the steady state solution for small and large diffusion rates.

Citation: Rong Zou, Shangjiang Guo. Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4189-4210. doi: 10.3934/dcdsb.2020093
References:
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H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

[2]

J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  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. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  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

[8]

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 J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

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Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

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S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011.  Google Scholar

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S. Guo, Patterns in a nonlocal time-delayed reaction-diffusion equation, Z. Angew. Math. Phys., 69 (2018), Art. 10, 31 pp. doi: 10.1007/s00033-017-0904-7.  Google Scholar

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S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 26 (2016), 545-580.  doi: 10.1007/s00332-016-9285-x.  Google Scholar

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X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[17]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

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X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

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K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  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 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

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L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

[23]

S. Li and S. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett., 101 (2020), 106066, 7 pp. doi: 10.1016/j.aml.2019.106066.  Google Scholar

[24]

S. LiJ. Wu and H. Nie, Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.  doi: 10.1016/j.camwa.2015.10.017.  Google Scholar

[25]

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

[26]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[27]

Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

[28]

L. Ma and S. Guo, Stability and bifurcation in a diffusive Lotka-Volterra system with delay, Comput. & Math. Appl., 72 (2016), 147-177.  doi: 10.1016/j.camwa.2016.04.049.  Google Scholar

[29]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York-London-Sydney, 1969.  Google Scholar

[30]

H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[32]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[33]

W. Sokol and J. A. Howell, Kinetics of phenol oxidation by washed cells, Biotechnology and Bioengineering, 23 (1981), 2039-2049.  doi: 10.1002/bit.260230909.  Google Scholar

[34]

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

[35]

B. Wang and Z. Zhang, Bifurcation analysis of a diffusive predator-prey model in spatially heterogeneous environment, Electron. J. Qual. Theory Differ. Equ., 42 (2017), 17 pp. doi: 10.14232/ejqtde.2017.1.42.  Google Scholar

[36]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018) 1559–1579. doi: 10.3934/dcdsb.2018059.  Google Scholar

[37]

S. Yan and S. Guo, Stability analysis of a stage structure model with spatiotemporal delay effect, Comput. Math. Appl., 73 (2017), 310-326.  doi: 10.1016/j.camwa.2016.11.029.  Google Scholar

[38]

R. Zou and S. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Comput. Math. Appl., 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

[2]

J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  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]

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

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[7]

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

[8]

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

[9]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, 2001. Google Scholar

[12]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011.  Google Scholar

[13]

S. Guo, Patterns in a nonlocal time-delayed reaction-diffusion equation, Z. Angew. Math. Phys., 69 (2018), Art. 10, 31 pp. doi: 10.1007/s00033-017-0904-7.  Google Scholar

[14]

S. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA J. Appl. Math., 82 (2017), 864-908.  doi: 10.1093/imamat/hxx018.  Google Scholar

[15]

S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 26 (2016), 545-580.  doi: 10.1007/s00332-016-9285-x.  Google Scholar

[16]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[17]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

[18]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[19]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[20]

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

[21]

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

[22]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

[23]

S. Li and S. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett., 101 (2020), 106066, 7 pp. doi: 10.1016/j.aml.2019.106066.  Google Scholar

[24]

S. LiJ. Wu and H. Nie, Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.  doi: 10.1016/j.camwa.2015.10.017.  Google Scholar

[25]

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

[26]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[27]

Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

[28]

L. Ma and S. Guo, Stability and bifurcation in a diffusive Lotka-Volterra system with delay, Comput. & Math. Appl., 72 (2016), 147-177.  doi: 10.1016/j.camwa.2016.04.049.  Google Scholar

[29]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York-London-Sydney, 1969.  Google Scholar

[30]

H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[32]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[33]

W. Sokol and J. A. Howell, Kinetics of phenol oxidation by washed cells, Biotechnology and Bioengineering, 23 (1981), 2039-2049.  doi: 10.1002/bit.260230909.  Google Scholar

[34]

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

[35]

B. Wang and Z. Zhang, Bifurcation analysis of a diffusive predator-prey model in spatially heterogeneous environment, Electron. J. Qual. Theory Differ. Equ., 42 (2017), 17 pp. doi: 10.14232/ejqtde.2017.1.42.  Google Scholar

[36]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018) 1559–1579. doi: 10.3934/dcdsb.2018059.  Google Scholar

[37]

S. Yan and S. Guo, Stability analysis of a stage structure model with spatiotemporal delay effect, Comput. Math. Appl., 73 (2017), 310-326.  doi: 10.1016/j.camwa.2016.11.029.  Google Scholar

[38]

R. Zou and S. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Comput. Math. Appl., 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.  Google Scholar

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