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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

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