November  2012, 17(8): 2745-2769. doi: 10.3934/dcdsb.2012.17.2745

On limit systems for some population models with cross-diffusion

1. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan

2. 

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

Received  April 2011 Revised  September 2011 Published  July 2012

This paper deals with the following reaction-diffusion system $$ (SP) \begin{equation} \left\{\begin{array}{11} \Delta[(1+\alpha v)u]+u(a-u-cv)=0, \\ \Delta[(1+\beta u)v]+v(b-du-v)=0, \end{array} \right. \end{equation} $$ in a bounded domain of $\Bbb{R}^N$ with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients $\alpha$ and $\beta$. For this purpose, our strategy is to study limiting behavior of positive solutions when $\alpha$ or $\beta$ goes to $\infty$ and derive the corresponding limit systems. We will obtain a priori estimates of $u$ and $v$ independently of $\beta$ (resp. $\alpha$) with small $\alpha\ge0$ (resp. $\beta\ge0$) in case $1\le N\le 3$ under Neumann boundary conditions, while we will obtain a priori estimates of $u$ and $v$ independently of $\alpha$ and $\beta$ in case $1\le N\le 5$ under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When $\alpha=0$ and $\beta\to\infty$, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.
Citation: Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745
References:
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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  Google Scholar

[2]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

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C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure. Appl. Math., 47 (1994), 1571-1594.  Google Scholar

[5]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  Google Scholar

[6]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.  Google Scholar

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K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[8]

K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066.  Google Scholar

[9]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  Google Scholar

[10]

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

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Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559.  Google Scholar

[12]

Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar

[13]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.  Google Scholar

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M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar

[15]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  Google Scholar

[16]

M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  Google Scholar

[17]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.  Google Scholar

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W.-M. Ni, Qualitative properties of solutions to elliptic systems, in "Stationary Partial Differential Equations. Vol. 1" (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.  Google Scholar

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A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001.  Google Scholar

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M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984.  Google Scholar

[21]

W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578. doi: 10.1006/jmaa.1996.0039.  Google Scholar

[22]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems, 9 (2003), 1049-1061.  Google Scholar

[23]

K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46-65. doi: 10.1016/S0022-247X(03)00162-8.  Google Scholar

[24]

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.  Google Scholar

[25]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[26]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, in "Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. VI" (ed. M. Chipot), Elsevier/North-Holland, Amsterdam, (2008), 411-501.  Google Scholar

show all references

References:
[1]

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

[2]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[3]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin-Heidelberg, 1998.  Google Scholar

[4]

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure. Appl. Math., 47 (1994), 1571-1594.  Google Scholar

[5]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  Google Scholar

[6]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.  Google Scholar

[7]

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[8]

K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066.  Google Scholar

[9]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  Google Scholar

[10]

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

[11]

Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559.  Google Scholar

[12]

Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar

[13]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.  Google Scholar

[14]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar

[15]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  Google Scholar

[16]

M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  Google Scholar

[17]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.  Google Scholar

[18]

W.-M. Ni, Qualitative properties of solutions to elliptic systems, in "Stationary Partial Differential Equations. Vol. 1" (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.  Google Scholar

[19]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001.  Google Scholar

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984.  Google Scholar

[21]

W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578. doi: 10.1006/jmaa.1996.0039.  Google Scholar

[22]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems, 9 (2003), 1049-1061.  Google Scholar

[23]

K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46-65. doi: 10.1016/S0022-247X(03)00162-8.  Google Scholar

[24]

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.  Google Scholar

[25]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[26]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, in "Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. VI" (ed. M. Chipot), Elsevier/North-Holland, Amsterdam, (2008), 411-501.  Google Scholar

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