# American Institute of Mathematical Sciences

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  November 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (11) : 4189-4210. doi: 10.3934/dcdsb.2020093
##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [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. [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. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [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. [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. [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. [8] Y. Du, R. 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. [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. [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. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, 2001. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [24] S. Li, J. 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. [25] A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, New York, 1925. [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. [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. [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. [29] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York-London-Sydney, 1969. [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. [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. [32] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, 41, American Mathematical Society, Providence, RI, 1995. [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. [34] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0. [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. [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. [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. [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.

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##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [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. [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. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [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. [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. [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. [8] Y. Du, R. 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. [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. [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. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, 2001. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [24] S. Li, J. 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. [25] A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, New York, 1925. [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. [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. [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. [29] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York-London-Sydney, 1969. [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. [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. [32] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, 41, American Mathematical Society, Providence, RI, 1995. [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. [34] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0. [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. [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. [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. [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.
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