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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

Kazuhiro Oeda 1,

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.
Keywords: protection zone, Prey-predator model, functional response., cross-diffusion, coexistence states.
Mathematics Subject Classification: Primary: 35B32, 35J57; Secondary: 92D2.
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), 13-75.  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), 719-730.  Google Scholar

[3]

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  Google Scholar

[4]

Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  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), 3932-3956.  Google Scholar

[6]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  Google Scholar

[7]

J. López-Gómez, "Spectral Theory and Nonlinear Functional Analysis," Research Notes in Mathematics, vol.426, Chapman and Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[8]

Y. Lou and W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  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), 193-203.  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), 3988-4009.  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), 501-520. Google Scholar

[12]

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  Google Scholar

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  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), 3933-3941.  Google Scholar

[15]

X. Zou and K. Wang, The protection zone of biological population, Nonlinear Anal. RWA, 12 (2011), 956-964.  Google Scholar

[16]

X. Zou and K. Wang, A robustness analysis of biological population models with protection zone, Applied Mathematical Modelling, 35 (2011), 5553-5563.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.  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), 719-730.  Google Scholar

[3]

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  Google Scholar

[4]

Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  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), 3932-3956.  Google Scholar

[6]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  Google Scholar

[7]

J. López-Gómez, "Spectral Theory and Nonlinear Functional Analysis," Research Notes in Mathematics, vol.426, Chapman and Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[8]

Y. Lou and W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  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), 193-203.  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), 3988-4009.  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), 501-520. Google Scholar

[12]

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  Google Scholar

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  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), 3933-3941.  Google Scholar

[15]

X. Zou and K. Wang, The protection zone of biological population, Nonlinear Anal. RWA, 12 (2011), 956-964.  Google Scholar

[16]

X. Zou and K. Wang, A robustness analysis of biological population models with protection zone, Applied Mathematical Modelling, 35 (2011), 5553-5563.  Google Scholar

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