Cooperative systems with any number of species
Authors:
Manuel Delgado and Antonio Suárez
Journal:
Quart. Appl. Math. 61 (2003), 683-699
MSC:
Primary 35R50; Secondary 35K57, 92D25
DOI:
https://doi.org/10.1090/qam/2019618
MathSciNet review:
MR2019618
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Abstract: In this paper we study the positive solutions of a cooperative system of any number of equations which consider the case of the slow diffusion and include the Lotka-Volterra model. We determine conditions of existence of global solution and blow-up in finite time in terms of the value of the spectral radius of a certain nonnegative matrix associated to the system. The results generalize the ones known for the particular case of two equations and we justify them by using the specific properties of nonnegative matrices which translate the cooperative character of the system.
F. Ayres Jr., Matrices, McGraw-Hill, México, 1969.
- Roberta Dal Passo and Piero de Mottoni, Some existence, uniqueness and stability results for a class of semilinear degenerate elliptic systems, Boll. Un. Mat. Ital. C (6) 3 (1984), no. 1, 203–231. MR 749291
- M. Delgado, J. López-Gómez, and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations 160 (2000), no. 1, 175–262. MR 1734533, DOI https://doi.org/10.1006/jdeq.1999.3655
- Manuel Delgado and Antonio Suárez, Stability and uniqueness for cooperative degenerate Lotka-Volterra model, Nonlinear Anal. 49 (2002), no. 6, Ser. A: Theory Methods, 757–778. MR 1894783, DOI https://doi.org/10.1016/S0362-546X%2801%2900138-9
M. Delgado and A. Suárez, Sistemas cooperativos de tipo Volterra-Lotka con tres especies, XVII Congress on Differential Equations and Applications/VII Congress on Applied Mathematics, Vol. I, II (Spanish) (Salamanca, 2001).
- Philip Korman, Dynamics of the Lotka-Volterra systems with diffusion, Appl. Anal. 44 (1992), no. 3-4, 191–207. MR 1284998, DOI https://doi.org/10.1080/00036819208840078
- Philip Korman and Anthony Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26 (1987), no. 2, 145–160. MR 921723, DOI https://doi.org/10.1080/00036818708839706
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F. Ayres Jr., Matrices, McGraw-Hill, México, 1969.
R. dal Passo and P. de Mottoni, Some existence, uniqueness and stability results for a class of semilinear degenerate elliptic systems, Boll. Un. Mat. Ital. C (6), 3, 203–231 (1975).
M. Delgado, J. López-Gómez, and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Eqns. 160, 175–262 (2000).
M. Delgado and A. Suárez, Stability and uniqueness for cooperative degenerate Lotka-Volterra model, Nonlinear Analysis, 49, 757–778 (2002).
M. Delgado and A. Suárez, Sistemas cooperativos de tipo Volterra-Lotka con tres especies, XVII Congress on Differential Equations and Applications/VII Congress on Applied Mathematics, Vol. I, II (Spanish) (Salamanca, 2001).
P. Korman, Dynamics of the Lotka-Volterra systems with diffusion, Appl. Anal. 44, 191–207 (1992).
P. Korman and A. Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26, 145–160 (1987).
Y. Lou, Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Nonlinear Analysis, 26, 1079–1095 (1996).
P. J. McKenna and W. Walter, On the Dirichlet problem for elliptic systems, Appl. Anal. 21, 207–224 (1986).
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
R. S. Varga, Matrix iterative analysis, Springer, Berlin, 2000.
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© Copyright 2003
American Mathematical Society
