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

Shanbing Li; Jianhua Wu

doi:10.3934/dcds.2017063

Article Contents

2017, Volume 37, Issue 3: 1539-1558. Doi: 10.3934/dcds.2017063

This issue Previous Article On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators Next Article Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth

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: March 31, 2016

Revised: September 30, 2016

Published: May 2017

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).

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary:35J65, 35B32;Secondary:92D25.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.