# American Institute of Mathematical Sciences

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 $$(SP) $$\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.$$$$ 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 and Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745
##### References:
 [1] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. [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. [3] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin-Heidelberg, 1998. [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. [5] Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. [6] Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. [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. [8] K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066. [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. [10] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. [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. [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. [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. [14] M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. [15] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. [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. [17] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [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. [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. [20] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. [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. [3] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin-Heidelberg, 1998. [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. [5] Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. [6] Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. [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. [8] K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066. [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. [10] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. [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. [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. [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. [14] M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. [15] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. [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. [17] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [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. [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. [20] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984. [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. [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. [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. [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. [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. [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.
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