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Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion. (English) Zbl 1074.35034
The paper is concerned with the existence and method of construction of solutions for a general class of strongly coupled elliptic systems with three classical types of reaction functions that correspond to competition, prey-predator, and cooperating models. Application of the method of upper and lower solutions and associated monotone iterations lead to some positive solutions of the competition system and to quasisolutions of the predator-prey and cooperating systems. Sufficient conditions for the existence of a unique positive solution for each model are given.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
35K57 Reaction-diffusion equations
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[1] Aronson, D.G.; Tesei, A.; Weinburger, H., A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. math. pure appl., 152, 259-280, (1988) · Zbl 0673.35054
[2] Boudiba, N.; Pierre, M., Global existence for coupled reaction – diffusion systems, J. math. anal. appl., 250, 1-12, (2000) · Zbl 0963.35077
[3] Deuring, P., An initial boundary value problem for a certain density-dependent diffusion system, Math. Z., 194, 375-396, (1987) · Zbl 0622.35038
[4] Kim, J.K., Smooth solutions for a quasilinear system of diffusion equations for a certain population model, Nonlinear anal. (TMA), 8, 1121-1144, (1984)
[5] Lou, Y.; Li, W.M., Diffusion, self-diffusion, and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[6] Lou, Y.; Ni, W.M.; Wu, Y., On the global existence of a cross-diffusion system, Discrete contin. dynam. syst., 4, 193-203, (1998) · Zbl 0960.35049
[7] Mimura, M., Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima math. J., 11, 621-635, (1981) · Zbl 0483.35045
[8] Mimura, M.; Kawasaki, K., Spatial segregation in competitive interaction-diffusion equations, J. math. biol., 9, 46-64, (1980) · Zbl 0425.92010
[9] Okubo, A., Diffusion and ecological problemsmathematical models, (1980), Springer Berlin
[10] Pao, C.V., Coexistence and stability of a competition diffusion system in population dynamics, J. math. anal. appl., 83, 54-76, (1981) · Zbl 0479.92013
[11] Pao, C.V., Numerical solutions for some coupled systems of nonlinear boundary value problems, Numer. math., 51, 381-394, (1987) · Zbl 0632.65111
[12] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044
[13] Pao, C.V., Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear anal. (TMA), 26, 1889-1903, (1996) · Zbl 0853.35056
[14] Pao, C.V., Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J. numer. anal., 36, 393-416, (1999) · Zbl 0921.65061
[15] Pozio, M.A.; Tesei, A., Global existence of solutions for a strongly coupled quasilinear parabolic system, Nonlinear anal. (TMA), 14, 657-689, (1990) · Zbl 0716.35034
[16] Redlinger, R., Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics, J. differential equations, 118, 219-252, (1995) · Zbl 0826.35054
[17] Ruan, W.H., Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. math. anal. appl., 197, 558-578, (1996) · Zbl 0855.35066
[18] Ruan, W.H., A competing reaction-diffusion system with small cross-diffusions, Can. appl. math. quart., 7, 69-91, (1999) · Zbl 0940.35076
[19] Ryu, K.; Ahn, I., Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. math. anal. appl., 283, 46-65, (2003) · Zbl 1115.35321
[20] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. theoret. biol., 79, 83-99, (1979)
[21] Shim, S.A., Uniform boundedness and convergence of solutions to the system with a single nonzero cross-diffusion, J. math. anal. appl., 279, 1-21, (2003) · Zbl 1032.35080
[22] Shim, S.A., Uniform boundedness and convergence of solutions to the system with cross-diffusions dominated by self-diffusion, Nonlinear anal. real world appl., 4, 65-86, (2003) · Zbl 1015.35020
[23] Y.P. Wu, Existence of stationary solutions for a class of cross-diffusion systems with small parameters, in: Lecture Notes on Contemporary Mathematics, Science Press, Beijing, to appear.
[24] Yagi, Y., Global solution to some quasilinear parabolic system in population dynamics, Nonlinear anal. (TMA), 21, 603-630, (1993) · Zbl 0810.35046
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