Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics

SIAM Journal on Mathematical Analysis

Volumn 29, Issue 6, 1998, Pages 1519-1536

  Kan On, Yukio ^a   Mimura, Masayasu ^b,c  

^a EHIME UNIVERSITY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

^c HIROSHIMA UNIVERSITY   (Japan)

Author keywords

Prey predator model; Singular perturbation; Stability

Indexed keywords

EID: 0032389543     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141097318328     Document Type: Article

References (19)

1. E. CONWAY, D. HOFF, AND J. SMOLLER, Large time behavior of solutions of systems of non-linear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), pp. 1-16.
2. W. A. COPPEL, Dichotomies in Stability Theory, Lecture Notes in Math. 629, Springer-Verlag, Berlin, 1978.
3. P. DE MOTTONI, Qualitative analysis for some quasi-linear parabolic systems, Inst. Math. Pol. Acad. Sci. Zam. 11/70, 190 (1979).
4. P. C. FIFE, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), pp. 497-521.
5. K. FUJII, Complexity-stability relationship of two-prey-one-predator species system model: Local and global stability, J. Theoret. Biol., 69 (1977), pp. 613-623.
6. S. B. Hsu, Predator-mediated coexistence and extinction, Math. Biosci., 54 (1981), pp. 231-248.
7. Y. KAN-ON, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), pp. 340-363.
8. Y. KAN-ON, Instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, in preparation.
9. Y. KAN-ON AND Q. FANG, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), pp. 343-349.
10. Y. KAN-ON AND E. YANAGIDA, Existence of non-constant stable equilibria in competition-diffusion equations. Hiroshima Math. J., 23 (1993), pp. 193-221.
11. K. KISHIMOTO AND H. F. WEINBERGER, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), pp. 15-21.
12. H. KOKUBU, Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5 (1988), pp. 455-501.
13. M. MATANO AND M. MIMURA, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), pp. 1049-1080.
14. M. MIMURA AND P. C. FIFE, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), pp. 189-207.
15. M. MIMURA AND Y. KAN-ON, Predation-mediated coexistence and segregation structures, in Patterns and Waves, T. Nishida et al., eds., Kinokuniya/North-Holland, Amsterdam, 1986, pp. 129-155.
16. Y. NISHIURA AND H. FUJII, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), pp. 1726-1770.
17. F. ROTHE, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math. 1072, Springer-Verlag, Berlin, New York, 1984.
18. Y. TAKEUCHI AND N. ADACHI, Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Biol., 45 (1983), pp. 877-900.
19. THE MATHEMATICAL SOCIETY OF JAPAN, Encyclopedic Dictionary of Mathematics, K. Ito, ed., MIT Press, Cambridge, MA, 1987.