A qualitative study on general Gause-type predator-prey models with constant diffusion rates

Journal of Mathematical Analysis and Applications

Volumn 344, Issue 1, 2008, Pages 217-230

  Ko, Wonlyul ^a   Ryu, Kimun ^b  

^a Korea University   (South Korea)

^b Chongju University   (South Korea)

Author keywords

Functional response; Hopf bifurcation; Locally globally asymptotically stable; Non constant positive solution; Persistence

Indexed keywords

EID: 42649116084     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2008.03.006     Document Type: Article

References (32)

1. p bounds of solutions of reaction-diffusion equations. Comm. Partial Differential Equations 4 (1979) 827-868
2. Blat J., and Brown K.J. Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal. 17 6 (1986) 1339-1353
3. Braun M. Differential Equations and Their Applications (1975), Springer-Verlag, Berlin
4. Casal A., Eilbeck J.C., and López-Gómez J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion. Differential Integral Equations 7 2 (1994) 411-439
5. Chen W., and Wang M. Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion. Math. Comput. Modelling 42 (2005) 31-44
6. Cheng K.S., Hsu S.B., and Lin S.S. Some results on global stability of a predator-prey system. J. Math. Biol. 12 (1981) 115-126
7. Crandall M.G., and Rabinowitz P.H. The Hopf bifurcation theorem in infinite dimensions. Arch. Ration. Mech. Anal. 67 1 (1977) 53-72
8. Du Y., and Lou Y. Some uniqueness and exact multiplicity results for a predator-prey model. Trans. Amer. Math. Soc. 349 6 (1997) 2443-2475
9. Du Y., and Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 2 (1998) 390-440
10. Du Y., and Lou Y. Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation. Proc. Roy. Soc. Edinburgh Sect. A 131 2 (2001) 321-349
11. Freedman H.I. Graphical stability, enrichment and pest control by a natural enemy. Math. Biosci. 31 (1976) 207-225
12. Freedman H.I. Deterministic Mathematical Models in Population Ecology. second ed. (1987), HIFR Consulting Ltd., Edmonton, Canada
13. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag, New York
14. Goel N.S., Maitra S.C., and Montroll E.W. On the Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43 (1971) 231-276
15. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. vol. 840 (1993), Springer-Verlag, Berlin
16. Holling C.S. The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 46 (1995) 1-60
17. Huang X.C., and Merrill S.J. Conditions for uniqueness of limit cycles in general predator-prey systems. Math. Biosci. 96 (1989) 47-60
18. Ko W., and Ryu K. Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J. Differential Equations 231 (2006) 534-550
19. Kuang Y., and Freedman H.I. Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biosci. 88 (1988) 67-84
20. Lin C.S., Ni W.M., and Takagi I. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988) 1-27
21. Liu Y. Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems. Proc. Amer. Math. Soc. 133 (2005) 3619-3626
22. May R.M. Stability and Complexity in Model Ecosystems (1974), Princeton Univ. Press, Princeton, NJ
23. Nirenberg L. Topics in Nonlinear Functional Analysis (1973), Courant Institute of Mathematical Science, New York
24. Oaten A., and Murdoch W.W. Functional response and stability in predator-prey system. Amer. Natur. 109 (1975) 289-298
25. Pang Y.H., and Wang M. Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion. Proc. London. Math. Soc. 88 (2004) 135-157
26. Pang Y.H., and Wang M. Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. Roy. Soc. Edinburgh Sect. A 131 (2003) 919-942
27. Peng R., and Wang M. On multiplicity and stability of positive solutions of a diffusive prey-predator model. J. Math. Anal. Appl. 316 (2006) 256-268
28. Rosenzweig M.L. Paradox of enrichment: Destabilization of exploitation ecosystem in ecological time. Science 171 (1971) 385-387
29. Rybakowski K.P. The Homotopy Index and Partial Differential Equations (1987), Springer-Verlag, Berlin
30. Smoller J. Shock Waves and Reaction-Diffusion Equations (1983), Springer-Verlag, New York
31. Sugie J., and Katayama M. Global asymptotic stability of a predator-prey system of Holling type. Nonlinear Anal. 38 (1999) 105-121
32. Ivlev V.S. Experimental Ecology of the Feeding of Fishes (1955), Yale Univ. Press