×

A Lotka-Volterra competition model with cross-diffusion. (English) Zbl 1276.92099

Summary: A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

MSC:

92D40 Ecology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lou, Y.; Ni, W.-M., Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131, 1, 79-131 (1996) · Zbl 0867.35032 · doi:10.1006/jdeq.1996.0157
[2] Ling, Z.; Pedersen, M., Coexistence of two species in a strongly coupled cooperative model, Mathematical and Computer Modelling, 45, 3-4, 371-377 (2007) · Zbl 1173.35711 · doi:10.1016/j.mcm.2006.05.011
[3] Lou, Y.; Ni, W.-M., Diffusion vs cross-diffusion: an elliptic approach, Journal of Differential Equations, 154, 1, 157-190 (1999) · Zbl 0934.35040 · doi:10.1006/jdeq.1998.3559
[4] Pao, C. V., Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Analysis. Theory, Methods & Applications A, 60, 7, 1197-1217 (2005) · Zbl 1074.35034 · doi:10.1016/j.na.2004.10.008
[5] Ko, W.; Ryu, K., On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species, Journal of Mathematical Analysis and Applications, 341, 2, 1133-1142 (2008) · Zbl 1160.35021 · doi:10.1016/j.jmaa.2007.11.018
[6] Leung, A., Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, Journal of Mathematical Analysis and Applications, 73, 1, 204-218 (1980) · Zbl 0427.35011 · doi:10.1016/0022-247X(80)90028-1
[7] Lakoš, N., Existence of steady-state solutions for a one-predator—two-prey system, SIAM Journal on Mathematical Analysis, 21, 3, 647-659 (1990) · Zbl 0705.92019 · doi:10.1137/0521034
[8] Li, L., Coexistence theorems of steady states for predator-prey interacting systems, Transactions of the American Mathematical Society, 305, 1, 143-166 (1988) · Zbl 0655.35021 · doi:10.2307/2001045
[9] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), New York, NY, USA: Plenum Press, New York, NY, USA · Zbl 0777.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.