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Diffusion, cross-diffusion and competitive interaction. (English) Zbl 1113.92064
Summary: The cross-diffusion competition systems were introduced by N. Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979); see also Lect. Notes Biomath. 54, 478–491 (1984; Zbl 0537.92028); J. Math. Biol. 9, 85–96 (1980; Zbl 0427.92015)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by a reaction-diffusion system which only includes a linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing instability of the corresponding reaction-diffusion system.

MSC:
92D40Ecology
35K57Reaction-diffusion equations
35B25Singular perturbations (PDE)
35B35Stability of solutions of PDE
35K55Nonlinear parabolic equations
References:
[1]Amann, H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math. vol. 133, pp. 9–126 (1993)
[2]Choi Y.S., Lui R., Yamada Y. (2003) Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discret. Contin. Dyn. Syst. 9, 1193–1200 · Zbl 1029.35116 · doi:10.3934/dcds.2003.9.1193
[3]Hirsch M.W. (1982) Differential equations and convergence almost everywhere of strongly monotone semiflows. PAM Technical Report, University of California, Berkeley
[4]Huang Y. (2005) How do cross-migration models arise? Math. Biosci. 195, 127–140 · Zbl 1065.92059 · doi:10.1016/j.mbs.2005.01.005
[5]Iida, M., Ninomiya, H. A reaction-diffusion approximation to a cross-diffusion system. In: Chipot, M., Ninomiya, H. (eds.) Recent Advances on Elliptic and Parabolic Issues. World Scientific, pp. 145–164 (2006)
[6]Kan-on Y. (1993) Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics. Hiroshima Math. J. 23, 509–536
[7]Kareiva P., Odell G. (1987) Swarms of predators exhibit ”preytaxis” if individual predator use area-restricted search. The Am. Nat. 130, 233–270 · doi:10.1086/284707
[8]Kishimoto K., Weinberger H.F. (1985) The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domains. J. Differ. Equ. 58, 15–21 · Zbl 0599.35080 · doi:10.1016/0022-0396(85)90020-8
[9]Lou Y., Ni W.-M. (1999) Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 · Zbl 0934.35040 · doi:10.1006/jdeq.1998.3559
[10]Lou Y., Ni W.-M., Yotsutani S. (2004) On a limiting system in the Lotka-Volterra competition with cross-diffusion. Discret. Contin. Dyn. Syst. 10, 435–458 · Zbl 1174.35360 · doi:10.3934/dcds.2004.10.435
[11]Lou Y., Ni W.-M., Wu Y. (1998) On the global existence of a cross-diffusion system. Discrete Contin. Dyn. Syst. 4, 193–203 · Zbl 0960.35049 · doi:10.3934/dcds.1998.4.193
[12]Matano H., Mimura M. (1983) Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. Kyoto Univ. 19, 1049–1079 · Zbl 0548.35063 · doi:10.2977/prims/1195182020
[13]Mimura M., Kawasaki K. (1980) Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9, 49–64 · Zbl 0425.92010 · doi:10.1007/BF00276035
[14]Mimura M., Nishiura Y., Tesei A., Tsujikawa T. (1984) Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14, 425–449
[15]Okubo A. (1980) Diffusion and Ecological Problems: Mathematical Models. Biomathematics. vol. 10, Springer, Berlin Heidelberg New York
[16]Shigesada N., Kawasaki K., Teramoto E. (1979) Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 · doi:10.1016/0022-5193(79)90258-3
[17]Turchin, P. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer (1998)
[18]Turing A.M. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 · doi:10.1098/rstb.1952.0012