Non-constant positive steady states of the Sel'kov model

Journal of Differential Equations

Volumn 190, Issue 2, 2003, Pages 600-620

  Wang, Mingxin ^a  

^a SOUTHEAST UNIVERSITY   (China)

Author keywords

Bifurcation; Global existence; Non constant positive steady states; Sel'kov model

Indexed keywords

EID: 0037845132     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-0396(02)00100-6     Document Type: Article

References (37)

1. P.N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980) 22-37.
2. K.J. Brown, F.A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal. TMA 24 (1995) 1713-1725.
3. J.B. Cameron, Spectral collocation and path-following methods for reaction-diffusion equations in one and two space dimensions, Ph.D. Thesis, Heriot Watt University, Edinburgh, 1994.
4. A. Casal, J.C. Eilbeck, J. Lópet-Gómez, Existence and uniqueness of coexistence states for a predator prey model with diffusion, Differential and Integral Equations 7 (1994) 411-439.
5. X.F. Chen, W.M. Ni, Y.W. Qi, M.X. Wang, Steady states of a strongly coupled prey-predator model, preprint.
6. E.N. Dancer, A counterexample of competing species equations, Differential Integral Equations 9 (1996) 239-246.
7. F.A. Davidson, B.P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh 130A (2000) 507-516.
8. M. Delgado, J. López-Gómez, A. Suarez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations 160 (2000) 175-262.
9. Y.H. Du, Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, preprint.
10. Y.H. Du, Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc. 349 (6) (1997) 2443-2475.
11. J.C. Eilbeck, Pattern formation and pattern selection in reaction-diffusion systems, in: B. Goodwin, P.T. Saunders (Eds.), Theoretical Biology: Epigenetic and Evolutionary Order, Edinburgh University Press, Edinburgh, 1989, pp. 31-41.
12. J.C. Eilbeck, J.E. Furter, Analysis of bifurcations in reaction-diffusion systems with no flux boundary conditions-the Sel'kov model, Proc. Roy. Soc. Edinburgh 125A (1995) 413-438.
13. B. Ermentrout, Strips or spots? Nonlinear effects in bifurcation of reaction diffusion equation on the square, Proc. Roy. Soc. London 434 (1991) 413-417.
14. J.E. Furter, J. López-Gómez, Diffusion mediated permanence problem for an heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh 127A (1997) 281-336.
15. Y.G. Gu, MX Wang, Existence of positive stationary solutions and threshold results for a reaction-diffusion system, J. Differential Equations 130 (2) (1996) 277-291.
16. Y.G. Gu, M.X. Wang, A semilinear parabolic system arising in the nuclear reactors, Chinese Sci. Bull. 39 (19) (1994) 1588-1592.
17. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Vol. 840 (3rd Printing), Springer, Berlin, 1993.
18. A. Hunding, P.G. Sorensen, Size adaptation of Turing prepatterns, J. Math. Biol. 26 (1988) 27-39.
19. Y. Kan-on, Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, J. Math. Anal. Appl. 243 (2000) 357-372.
20. Y. Kan-on, M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal. 29 (1998) 1519-1536.
21. L. Li, Coexistence theorems of steady-states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988) 143-166.
22. C.S. Lin, W.M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotais systems, J., Differential Equations 72 (1988) 1-27.
23. J. López-Gómez, J.C. Eilbeck, M. Molina, K.N. Duncan, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal. 12 (1992) 405-428.
24. J. López-Gómez, R. Pardo San Gil, Coexistence in a simple food chain with diffusion, J. Math. Biol. 30 (1992) 655-668.
25. Y. Lou, S. Martinez, W.M. Ni, On 3 × 3 Lotka-Volterra competition systems with cross-diffusion, Discrete Continuous Dyn. Systems 6 (1) (2000) 175-190.
26. Y. Lou, W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996) 79-131.
27. Y. Lou, W.M. Ni, Diffusion, vs. cross-diffusion: an elliptic approach, J. Differential Equations 154 (1999) 157-190.
28. M. Mimura, Y. Nishiura, A. Tesei, T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J. 14 (1984) 425-496.
29. P. de Motoni, F. Rothe, Convergence to homogeneous equilibrium State for a generalized Volterra-Lotka systems, SIAM J. Appl. Math. 37 (1979) 648-663.
30. J.D. Murry, Mathematical Biology, 2nd, Corrected Edition, Springer, Berlin, 1993.
31. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York, 1973-1974.
32. P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487-513.
33. S.D. Scott, P. Gray, Chemical reactions in isothermal systems: oscillations and instabilities, in: S. Sarkar (Ed.), Nonlinear Phenomena and Chaos, Adam Hilger, Bristol, 1986, pp. 70-96.
34. E.E. Sel'kov, Self-oscillations in glycolysis, European J. Biochem. 74 (1968) 79-86.
35. J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Springer, Berlin, 1994.
36. A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. 237B (1952) 37-72.
37. M.X. Wang, Stationary patterns of strongly coupled prey-predator models, preprint.