The linear and nonlinear diffusion of the competitive Lotka-Volterra model. (English) Zbl 1129.34037
Competitive Lotka-Volterra systems with a linear or nonlinear diffusion term are studied. Depending on the parameter values, these systems have several nonnegative equilibria, at which one of the species does not extinct. In two cases, sufficient conditions are given for the existence of a unique positive equilibrium at which all species are persistent, and for its global asymptotic stability.
Reviewer: Kazuyuki Yagasaki (Gifu)
MSC:
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D25 | Population dynamics (general) |
Keywords:
competitive Lotka-Volterra model; diffusion; positive equilibrium; global asymptotic stabilityReferences:
| [1] | Freedman, H. I.; Waltman, P., Mathematical models of population interaction with dispersal I: Stability of habitats with and without a predator, SIAM J. Appl. Math., 32, 631-648 (1977) · Zbl 0362.92006 |
| [2] | Freedman, H. I.; Rai, B.; Waltman, P., Mathematical models of population interaction with dispersal II: Differential survival in a change of habitat, J. Math. Anal. Appl., 115, 140-154 (1986) · Zbl 0588.92020 |
| [3] | Lu, Z.; Takeuchi, Y., Global asymptotic behaviour in single-species discrete diffusion systems, J. Math. Biol., 32, 67-77 (1993) · Zbl 0799.92014 |
| [4] | Cui, J.; Chen, L., The effect of diffusion on the time varying Logistic population growth, Comput. Math. Appl., 36, 1-9 (1998) · Zbl 0934.92025 |
| [5] | Song, X.; Chen, L., Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay, Comput. Math. Appl., 35, 33-40 (1998) · Zbl 0903.92029 |
| [6] | Takeuchi, Y., Global Dynamical Properties of Lotka-Volterra Systems (1996), World Scientific: World Scientific Singapore · Zbl 0844.34006 |
| [7] | Levin, S. A., Spatial patterning and structure of ecological communities, (Some Mathematical Questions in Biology VII (1976), American Mathematical Society: American Mathematical Society Providence) · Zbl 0338.92017 |
| [8] | Allen, L. J., Persistence and extinction in single-species reaction-diffusion models, Bull. Math. Biol., 45, 209-227 (1983) · Zbl 0543.92020 |
| [9] | Takeuchi, Y., Conflict between the need to forage and the need to avoid competition: Persistence of two-species models, Math. Biosci., 120, 181-194 (1990) · Zbl 0703.92024 |
| [10] | Chen, L.; Chen, J., Nonlinear Biological Dynamic Systems (1993), Science Press: Science Press Beijing, (in Chinese) |
| [11] | Chen, L., Mathematical Models and Methods in Ecology (1988), Science Press: Science Press Beijing, (in Chinese) |
| [12] | Guckenheimer, J.; Holmes, P., Nonlinear Oscillation, Dynamical Systems and Bifurcation of Vector Fields (1983), Springer-Verlag · Zbl 0515.34001 |
| [13] | Kuang, Y.; Takeuchi, Y., Predator-prey dynamic in models of prey dispersal in two patch environments, Math. Biosci., 120, 77-98 (1994) · Zbl 0793.92014 |
| [14] | Hirsch, M. W., Systems of differential equations which are competitive and cooperative I: Limit sets, SIAM J. Math. Anal., 13, 167-179 (1982) · Zbl 0494.34017 |
| [15] | Smith, H., Periodic orbits of competitive and cooperative systems, J. Differential Equations, 65, 361-373 (1986) · Zbl 0615.34027 |
| [16] | Hale, J. K., Ordinary Differential Equations (1969), Wiley: Wiley New York · Zbl 0186.40901 |
| [17] | Ruan, S.; Wolkowicz, G., Bifurcation analysis of a chemostate model with a distributed delay, J. Math. Anal. Appl., 204, 786-812 (1996) · Zbl 0879.92031 |
| [18] | Gonzeles, E. A., Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc., 143, 201-222 (1969) · Zbl 0187.34401 |
| [19] | Ye, Y., (Theory of Limit Cycles. Theory of Limit Cycles, Translations of Mathematical Monographs, vol. 66 (1986), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0588.34022 |
| [20] | Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102 |
| [21] | Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602 |
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