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2013, 2013(special): 597-603. doi: 10.3934/proc.2013.2013.597

Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response

1. 

Department of Applied Mathematics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received  September 2012 Revised  March 2013 Published  November 2013

This paper is concerned with the steady state problem of a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. A sufficient condition for the existence of positive steady state solutions is given. Our proof is based on the bifurcation theory and some a priori estimates.
Citation: Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar

[2]

Y.S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 719.   Google Scholar

[3]

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.   Google Scholar

[4]

Y. Du and X. Liang, A diffusive competition model with a protection zone,, J. Differential Equations, 244 (2008), 61.   Google Scholar

[5]

Y. 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.   Google Scholar

[6]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone,, J. Differential Equations, 229 (2006), 63.   Google Scholar

[7]

J. López-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Research Notes in Mathematics, (2001).   Google Scholar

[8]

Y. Lou and W.M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.   Google Scholar

[9]

Y. Lou, W.M. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193.   Google Scholar

[10]

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.   Google Scholar

[11]

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

[12]

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487.   Google Scholar

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83.   Google Scholar

[14]

P.V. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933.   Google Scholar

[15]

X. Zou and K. Wang, The protection zone of biological population,, Nonlinear Anal. RWA, 12 (2011), 956.   Google Scholar

[16]

X. Zou and K. Wang, A robustness analysis of biological population models with protection zone,, Applied Mathematical Modelling, 35 (2011), 5553.   Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar

[2]

Y.S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 719.   Google Scholar

[3]

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.   Google Scholar

[4]

Y. Du and X. Liang, A diffusive competition model with a protection zone,, J. Differential Equations, 244 (2008), 61.   Google Scholar

[5]

Y. 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.   Google Scholar

[6]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone,, J. Differential Equations, 229 (2006), 63.   Google Scholar

[7]

J. López-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Research Notes in Mathematics, (2001).   Google Scholar

[8]

Y. Lou and W.M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.   Google Scholar

[9]

Y. Lou, W.M. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193.   Google Scholar

[10]

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.   Google Scholar

[11]

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

[12]

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487.   Google Scholar

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83.   Google Scholar

[14]

P.V. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933.   Google Scholar

[15]

X. Zou and K. Wang, The protection zone of biological population,, Nonlinear Anal. RWA, 12 (2011), 956.   Google Scholar

[16]

X. Zou and K. Wang, A robustness analysis of biological population models with protection zone,, Applied Mathematical Modelling, 35 (2011), 5553.   Google Scholar

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