zbMATH — the first resource for mathematics

Coexistence of two species in a strongly coupled cooperative model. (English) Zbl 1173.35711
Summary: The cooperative two-species Lotka-Volterra model is discussed. We study the existence of solutions to a elliptic system with homogeneous Dirichlet boundary conditions. Our results show that this problem possesses at least one coexistence state if the birth rates are big and self-diffusions and the intra-specific competitions are strong.

35Q80 Applications of PDE in areas other than physics (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
92D25 Population dynamics (general)
Full Text: DOI
[1] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. theoret. biol., 79, 83-99, (1979)
[2] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[3] Lou, Y.; Ni, W.M., Diffusion, vs. cross-diffusion: an elliptic approach, J. differential equations, 154, 157-190, (1999) · Zbl 0934.35040
[4] Lou, Y.; Thomas, N.; Ni, W.M., On diffusion-induced blowups in a mutualistic model, Nonlinear anal., 45, 329-342, (2001) · Zbl 0980.35059
[5] Lou, Y., Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Nonlinear anal., 26, 1079-1095, (1996) · Zbl 0856.35038
[6] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer-Verlag Berlin, New York · Zbl 0422.92025
[7] 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
[8] Ahn, I.; Li, L., Positive solutions of certain elliptic systems with density dependent diffusions, Proc. roy. soc. Edinburgh sect. A, 125, 1031-1050, (1995) · Zbl 0840.35032
[9] Chen, B.; Peng, R., Coexistence states of a strongly coupled prey – predator model, J. partial differential equations, 18, 154-166, (2005) · Zbl 1330.35466
[10] Pao, C.V., Strongly coupled elliptic systems and applications to lotka – volterra models with cross-diffusion, Nonlinear anal., 60, 1197-1217, (2005) · Zbl 1074.35034
[11] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044
[12] Pedersen, M.; Lin, Z.G., Stationary patterns in one-predator, two-prey models, Differential integral equations, 14, 605-612, (2001) · Zbl 1161.35434
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.