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March  2017, 37(3): 1539-1558. doi: 10.3934/dcds.2017063

Effect of cross-diffusion in the diffusion prey-predator model with a protection zone

Shanbing Li 1, and  Jianhua Wu 2,

1. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China
2. 

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

Received  April 2016 Revised  October 2016 Published  December 2016

Fund Project: The work is supported by the Natural Science Foundation of China (11271236,11401356,11671243,61672021), the Natural Science Basic Research Plan in Shaanxi Province of China (No.2015JQ1023), the Shaanxi New-star Plan of Science and Technology (No.2015KJXX-21).

In this work, we continue the mathematical study started in [K. Oeda, J. Differential Equations 250 (2011) 3988-4009] on the analytic aspects of the diffusion prey-predator system with a protection zone and cross-diffusion. For small birth rates of two species and large cross-diffusion for the prey, the detailed structure of positive solutions is established by the bifurcation theory and the Lyapunov-Schmidt reduction, which is determined by a finite dimensional limiting system. Moreover, we prove that the stability of positive solutions changes only at every turning point by a spectral analysis for the linearized eigenvalue problem of the limiting system and its perturbation.

Keywords: Cross-diffusion, protection zone, Lyapunov-Schmidt reduction, stability.
Mathematics Subject Classification: Primary:35J65, 35B32;Secondary:92D25.
Citation: Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063
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.

[2]

Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.

[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.

[4]

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.

[5]

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.

[6]

X. He and S. N. Zheng, Protection zone in a modified Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2027-2038.  doi: 10.3934/dcdsb.2015.20.2027.

[7]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, preprint, arXiv: 1505.06625.

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, New York, 1966.

[9]

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.  doi: 10.1016/j.jde.2003.08.003.

[10]

K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 10 (2009), 943-965.  doi: 10.1016/j.nonrwa.2007.11.015.

[11]

K. Kuto, Stability and Hopf bifurcation of steady-state solutions to an SKT model in a spatially heterogeneous environment, Discrete Contin. Dynam. Syst., 24 (2009), 489-509.  doi: 10.3934/dcds.2009.24.489.

[12]

S. B. Li, J. H. Wu, S. Y. Liu and Y. Y. Dong, Effects of cross-diffusion and protection zone in the Leslie-Gower predator-prey model, submitted.

[13]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis. Research Notes in Mathematics vol. 426, CRC Press, Boca Raton, FL, 2001. doi: 10.1201/9781420035506.

[14]

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.

[15]

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

[16]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[17]

Y. X. Wang and W. T. Li, Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system, Nonlinear Anal. Real World Appl., 14 (2013), 1235-1246.  doi: 10.1016/j.nonrwa.2012.09.015.

[18]

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.

[19]

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

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.

[2]

Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.

[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.

[4]

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.

[5]

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.

[6]

X. He and S. N. Zheng, Protection zone in a modified Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2027-2038.  doi: 10.3934/dcdsb.2015.20.2027.

[7]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, preprint, arXiv: 1505.06625.

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, New York, 1966.

[9]

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.  doi: 10.1016/j.jde.2003.08.003.

[10]

K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 10 (2009), 943-965.  doi: 10.1016/j.nonrwa.2007.11.015.

[11]

K. Kuto, Stability and Hopf bifurcation of steady-state solutions to an SKT model in a spatially heterogeneous environment, Discrete Contin. Dynam. Syst., 24 (2009), 489-509.  doi: 10.3934/dcds.2009.24.489.

[12]

S. B. Li, J. H. Wu, S. Y. Liu and Y. Y. Dong, Effects of cross-diffusion and protection zone in the Leslie-Gower predator-prey model, submitted.

[13]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis. Research Notes in Mathematics vol. 426, CRC Press, Boca Raton, FL, 2001. doi: 10.1201/9781420035506.

[14]

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.

[15]

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

[16]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[17]

Y. X. Wang and W. T. Li, Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system, Nonlinear Anal. Real World Appl., 14 (2013), 1235-1246.  doi: 10.1016/j.nonrwa.2012.09.015.

[18]

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.

[19]

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

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