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.
Citation: |
[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.
![]() ![]() |