American Institute of Mathematical Sciences

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

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

Yaying Dong 1, , Shanbing Li 2, and  Yanling Li 3,

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.

Keywords: Predator-prey model, cross-diffusion, protection zone, positive stationary solutions, limiting behavior.
Mathematics Subject Classification: Primary: 35J55, 92D40; Secondary: 35B30, 35B32.
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. Cui, J. 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. Du, R. 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. Li, S. Y. Liu, J. 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. Li, J. 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. Li, J. 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. Zeng, W. 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. Cui, J. 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. Du, R. 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. Li, S. Y. Liu, J. 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. Li, J. 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. Li, J. 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. Zeng, W. 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]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
[2]

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
[3]

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
[4]

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
[5]

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
[6]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[7]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[8]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[9]

Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051
[10]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[11]

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
[12]

Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020
[13]

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
[14]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[15]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[16]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495
[17]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294
[18]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[19]

El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355
[20]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (114)
  • HTML views (197)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences