• Previous Article
    Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation
  • CPAA Home
  • This Issue
  • Next Article
    Global existence and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model
September  2019, 18(5): 2511-2528. doi: 10.3934/cpaa.2019114

Effects of dispersal for a predator-prey model in a heterogeneous environment

1. 

School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, China

2. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China

3. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China

Received  June 2018 Revised  November 2018 Published  April 2019

Fund Project: The work is supported by the Natural Science Foundation of China (11801431, 61672021), the Postdoctoral Science Foundation of China (2018T111014, 2018M631133), the Natural Science Foundation of Shaanxi Province (2018JQ1004, 2018JQ1017), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 18JK0343)

In this paper, we study the stationary problem of a predator-prey cross-diffusion system with a protection zone for the prey. We first apply the bifurcation theory to establish the existence of positive stationary solutions. Furthermore, as the cross-diffusion coefficient goes to infinity, the limiting behavior of positive stationary solutions is discussed. These results implies that the large cross-diffusion has beneficial effects on the coexistence of two species. Finally, we analyze the limiting behavior of positive stationary solutions as the intrinsic growth rate of the predator species goes to infinity.

Citation: Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114
References:
[1]

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

[2]

R. H. CuiJ. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Differential Equations, 256 (2014), 108-129. doi: 10.1016/j.jde.2013.08.015. Google Scholar

[3]

Y. H. Du and J. P. 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

[4]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557–4593.Google Scholar

[5]

Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005. Google Scholar

[6]

Y. H. DuR. Peng and M. X. 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

[7]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257. doi: 10.1007/s00285-016-1082-5. Google Scholar

[8]

S. B. LiS. Y. LiuJ. H. Wu and Y. Y. Dong, Positive solutions for Lotka-Volterra competition system with large cross-diffusion in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 36 (2017), 1-19. doi: 10.1016/j.nonrwa.2016.12.004. Google Scholar

[9]

S. B. Li and J. H. Wu, Effect of cross-diffusion in the diffusion prey-predator model with a protection zone, Discrete Contin. Dynam. Syst., 37 (2017), 411-430. doi: 10.3934/dcds.2017063. Google Scholar

[10]

S. B. Li, J. H. Wu and S. Y. Liu, Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone, Calc. Var. Partial Differential Equations, (2017), 56–82. doi: 10.1007/s00526-017-1159-z. Google Scholar

[11]

S. B. Li and Y. Yamada, Effect of cross-diffusion in the diffusion prey-predator model with a protection zone Ⅱ, J. Math. Anal. Appl., 461 (2018), 971-992. doi: 10.1016/j.jmaa.2017.12.029. Google Scholar

[12]

S. B. Li and J. H. Wu, Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system, J. Differential Equations, 265 (2018), 3754-3791. doi: 10.1016/j.jde.2018.05.017. Google Scholar

[13]

S. B. LiJ. H. Wu and Y. Y. Dong, Effects of a degeneracy in a diffusive predator-prey model with Holling Ⅱ functional response, Nonlinear Anal. Real World Appl., 43 (2018), 78-95. doi: 10.1016/j.nonrwa.2018.02.003. Google Scholar

[14]

S. B. LiJ. H. Wu and Y. Y. Dong, Effects of degeneracy and response function in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483. doi: 10.1088/1361-6544/aaa2de. Google Scholar

[15]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400–1406.Google Scholar

[16]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001.Google Scholar

[17]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009. doi: 10.1016/j.jde.2011.01.026. Google Scholar

[18]

K. Oeda, Coexistence states of a prey-predator model with cross-diffusion and a protection zone, Adv. Math. Sci. Appl., 22 (2012), 501-520. Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, J. Funct. Anal., 7 (1971) 487–513.Google Scholar

[20]

Y. X. Wang and W. T. Li, Effects of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. Real World Appl., 14 (2013), 224-245. doi: 10.1016/j.nonrwa.2012.06.001. Google Scholar

[21]

Y. X. Wang and W. T. Li, Uniqueness and global stability of positive stationary solution for a predator-prey system, J. Math. Anal. Appl., 462 (2018), 577-589. doi: 10.1016/j.jmaa.2018.02.032. Google Scholar

[22]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Beijing: Science Press, 1990.Google Scholar

[23]

X. Z. ZengW. T. Zeng and L. Y. Liu, Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626. doi: 10.1016/j.jmaa.2018.02.060. Google Scholar

show all references

References:
[1]

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

[2]

R. H. CuiJ. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Differential Equations, 256 (2014), 108-129. doi: 10.1016/j.jde.2013.08.015. Google Scholar

[3]

Y. H. Du and J. P. 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

[4]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557–4593.Google Scholar

[5]

Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005. Google Scholar

[6]

Y. H. DuR. Peng and M. X. 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

[7]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257. doi: 10.1007/s00285-016-1082-5. Google Scholar

[8]

S. B. LiS. Y. LiuJ. H. Wu and Y. Y. Dong, Positive solutions for Lotka-Volterra competition system with large cross-diffusion in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 36 (2017), 1-19. doi: 10.1016/j.nonrwa.2016.12.004. Google Scholar

[9]

S. B. Li and J. H. Wu, Effect of cross-diffusion in the diffusion prey-predator model with a protection zone, Discrete Contin. Dynam. Syst., 37 (2017), 411-430. doi: 10.3934/dcds.2017063. Google Scholar

[10]

S. B. Li, J. H. Wu and S. Y. Liu, Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone, Calc. Var. Partial Differential Equations, (2017), 56–82. doi: 10.1007/s00526-017-1159-z. Google Scholar

[11]

S. B. Li and Y. Yamada, Effect of cross-diffusion in the diffusion prey-predator model with a protection zone Ⅱ, J. Math. Anal. Appl., 461 (2018), 971-992. doi: 10.1016/j.jmaa.2017.12.029. Google Scholar

[12]

S. B. Li and J. H. Wu, Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system, J. Differential Equations, 265 (2018), 3754-3791. doi: 10.1016/j.jde.2018.05.017. Google Scholar

[13]

S. B. LiJ. H. Wu and Y. Y. Dong, Effects of a degeneracy in a diffusive predator-prey model with Holling Ⅱ functional response, Nonlinear Anal. Real World Appl., 43 (2018), 78-95. doi: 10.1016/j.nonrwa.2018.02.003. Google Scholar

[14]

S. B. LiJ. H. Wu and Y. Y. Dong, Effects of degeneracy and response function in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483. doi: 10.1088/1361-6544/aaa2de. Google Scholar

[15]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400–1406.Google Scholar

[16]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001.Google Scholar

[17]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009. doi: 10.1016/j.jde.2011.01.026. Google Scholar

[18]

K. Oeda, Coexistence states of a prey-predator model with cross-diffusion and a protection zone, Adv. Math. Sci. Appl., 22 (2012), 501-520. Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, J. Funct. Anal., 7 (1971) 487–513.Google Scholar

[20]

Y. X. Wang and W. T. Li, Effects of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. Real World Appl., 14 (2013), 224-245. doi: 10.1016/j.nonrwa.2012.06.001. Google Scholar

[21]

Y. X. Wang and W. T. Li, Uniqueness and global stability of positive stationary solution for a predator-prey system, J. Math. Anal. Appl., 462 (2018), 577-589. doi: 10.1016/j.jmaa.2018.02.032. Google Scholar

[22]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Beijing: Science Press, 1990.Google Scholar

[23]

X. Z. ZengW. T. Zeng and L. Y. Liu, Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626. doi: 10.1016/j.jmaa.2018.02.060. Google Scholar

[1]

Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063

[2]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597

[3]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875

[4]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[5]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189

[6]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[7]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[8]

Daniel Ryan, Robert Stephen Cantrell. Avoidance behavior in intraguild predation communities: A cross-diffusion model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1641-1663. doi: 10.3934/dcds.2015.35.1641

[9]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209

[10]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435

[11]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026

[12]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239

[13]

Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073

[14]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203

[15]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[16]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145

[17]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501

[18]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

[19]

Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423

[20]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (30)
  • HTML views (145)
  • Cited by (0)

Other articles
by authors

[Back to Top]