-
Previous Article
A note on global stability in the periodic logistic map
- DCDS-B Home
- This Issue
-
Next Article
Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model
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 |
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.
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. 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. |
[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. 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. |
[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. |
show all references
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. 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. |
[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. 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. |
[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. |
[1] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[2] |
Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148 |
[3] |
Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263 |
[4] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[5] |
Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021007 |
[6] |
Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020468 |
[7] |
Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328 |
[8] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[9] |
Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 |
[10] |
Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031 |
[11] |
Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 |
[12] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[13] |
Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 |
[14] |
Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 |
[15] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[16] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[17] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[18] |
Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020054 |
[19] |
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 |
[20] |
Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]