Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

Kuto, Kousuke

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1943-1959

This paper studies the asymptotic behavior of coexistence steady-states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni [18] derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on [10] implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This paper gives a rigorous proof of his observation as far as the stationary problem. As a key ingredient of the proof, we establish a uniform L∞ estimate for all steady-states. Thanks to this a priori estimate, we show that the asymptotic profile of coexistence steady-states can be characterized by a solution of the limiting system.

Classification : 35B45, 35B50, 35B32, 35J57, 92D25
Keywords: Cross-diffusion, Nonlinear elliptic system, A priori estimate, Maximum principle, Limiting system, Bifurcation

[1] Crandall, M.G.; Rabinowitz, P.H. Bifurcation from simple eigenvalues, J. Funct. Anal., Volume 8 (1971), pp. 321-340 | MR | Zbl | DOI

[2] Dancer, E.N.; Du, Y. Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differ. Equ., Volume 114 (1994), pp. 434-475 | MR | Zbl | DOI

[3] Dancer, E.N.; Hilhorst, D.; Mimura, M.; Peletier, L.A. Spatial segregation limit of a competition-diffusion system, Eur. J. Appl. Math., Volume 10 (1999), pp. 97-115 | MR | Zbl | DOI

[4] Dancer, E.N.; Zhang, Z. Dynamics of Lotka-Volterra competition systems with large interaction, J. Differ. Equ., Volume 182 (2002), pp. 470-489 | MR | Zbl | DOI

[5] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg, 1998 | MR | Zbl

[6] Hirose, T.; Yamada, Y. Multiple existence of positive solutions of competing species equations with diffusion and large interactions, Adv. Math. Sci. Appl., Volume 12 (2002), pp. 435-453 | MR | Zbl

[7] Jüngel, A. Diffusive and nondiffusive population models (Naldi, G.; Pareschi, L.; Toscani, G., eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser, Basel, 2010, pp. 397-425 | Zbl | MR | DOI

[8] Kan-on, Y. Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., Volume 28 (1993), pp. 509-536 | MR | Zbl

[9] Kan-on, Y. On the structure of positive solutions for the Shigesada-Kawasaki-Teramoto model with large interspecific competition rate, Int. J. Bifurc. Chaos Appl. Sci. Eng., Volume 30 (2020) no. 1 | MR | DOI

[10] Kan-on, Y. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates, Discrete Contin. Dyn. Syst., Volume 40 (2020), pp. 3561-3570 | MR | DOI

[11] Kolokolnikov, T.; Wei, J. Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., Volume 71 (2011), pp. 1428-1457 | MR | Zbl | DOI

[12] Krasnoselskii, M. Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1964 | Zbl

[13] Kuto, K. Limiting structure of shrinking solutions to the stationary SKT model with large cross-diffusion, SIAM J. Math. Anal., Volume 47 (2015), pp. 3993-4024 | MR | DOI

[14] Kuto, K.; Yamada, Y. Positive solutions for Lotka-Volterra competition systems with large cross-diffusion, Appl. Anal., Volume 89 (2010), pp. 1037-1066 | MR | Zbl | DOI

[15] Kuto, K.; Yamada, Y. On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. Syst., Ser. B, Volume 17 (2012), pp. 2745-2769 | MR | Zbl | DOI

[16] Li, Q.; Wu, Y. Stability analysis on a type of steady state for the SKT competition model with large cross diffusion, J. Math. Anal. Appl., Volume 462 (2018), pp. 1048-1078 | MR | DOI

[17] Lou, Y.; Ni, W.-M. Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ., Volume 131 (1996), pp. 79-131 | MR | Zbl | DOI

[18] Lou, Y.; Ni, W.-M. Diffusion vs cross-diffusion: an elliptic approach, J. Differ. Equ., Volume 154 (1999), pp. 157-190 | MR | Zbl | DOI

[19] Lou, Y.; Ni, W.-M.; Yotsutani, S. On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., Volume 10 (2004), pp. 435-458 | MR | Zbl | DOI

[20] Lou, Y.; Ni, W.-M.; Yotsutani, S. Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., Volume 35 (2015), pp. 1589-1607 | MR | Zbl | DOI

[21] Mimura, M. Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., Volume 11 (1981), pp. 621-635 | MR | Zbl | DOI

[22] Mimura, M.; Kawasaki, K. Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., Volume 9 (1980), pp. 49-64 | MR | Zbl | DOI

[23] Mimura, M.; Nishiura, Y.; Tesei, A.; Tsujikawa, T. Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., Volume 14 (1984), pp. 425-449 | MR | Zbl | DOI

[24] Mori, T.; Suzuki, T.; Yotsutani, S. Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation, Math. Models Methods Appl. Sci., Volume 11 (2018), pp. 2191-2210 | MR | DOI

[25] Ni, W.-M. The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 82, SIAM, Philadelphia, 2011 | MR | Zbl

[26] Ni, W.-M.; Wu, Y.; Xu, Q. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion, Discrete Contin. Dyn. Syst., Volume 34 (2014), pp. 5271-5298 | MR | Zbl | DOI

[27] Okubo, A.; Levin, L.A. Diffusion and Ecological Problems: Modern Perspective, Interdisciplinary Applied Mathematics, vol. 14, Springer-Verlag, New York, 2001 | MR | Zbl

[28] Shigesada, N.; Kawasaki, K.; Teramoto, E. Spatial segregation of interacting species, J. Theor. Biol., Volume 79 (1979), pp. 83-99 | MR | DOI

[29] Wang, L.; Wu, Y.; Xu, Q. Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., Volume 159 (2017), pp. 424-457 | MR | DOI

[30] Wu, Y. The instability of spiky steady states for a competing species model with cross-diffusion, J. Differ. Equ., Volume 213 (2005), pp. 289-340 | MR | Zbl | DOI

[31] Wu, Y.; Xu, Q. The existence and structure of large spiky steady states for S-K-T competition systems with cross diffusion, Discrete Contin. Dyn. Syst., Volume 29 (2011), pp. 367-385 | MR | Zbl | DOI

[32] Yamada, Y. Positive solutions for Lotka-Volterra systems with cross-diffusion (Chipot, M., ed.), Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 6, Elsevier, Amsterdam, 2008, pp. 411-501 | MR | Zbl | DOI

[33] Yamada, Y. Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion (Du, Y.; Ishii, H.; Lin, W.-Y., eds.), Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, 2009, pp. 282-299 | MR | Zbl | DOI

[34] Yotsutani, S. Multiplicity of solutions in the Lotka-Volterra competition with cross-diffusion, RIMS Kokyuroku, Volume 1838 (2013), pp. 116-125