Global asymptotic stability of a ratio-dependent predator-prey system with diffusion. (English) Zbl 1093.35039

The authors investigate a predator-prey system with diffusion. The main attention is paid to the global asymptotic stability of the positive constant equilibrium of the system. In order to prove the main results, the authors consider the local asymptotic stability and then obtain some estimation of the solutions. The proof is completed by constructing Lyapunov functions, the method of upper and lower solutions combined with the monotone iteration.


35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
92D25 Population dynamics (general)
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[1] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, OIKOS, 60, 69-75 (1991)
[2] Barbălat, I., Systemes d’equations differentielle d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959)
[3] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal. TMA, 32, 381-408 (1998) · Zbl 0946.34061
[4] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 1530-1535 (1992)
[5] Fan, Y. H.; Li, W. T.; Wang, L. L., Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional response, Nonlinear Anal. RWA, 5, 247-263 (2004) · Zbl 1069.34098
[6] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[7] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0752.34039
[8] Hanski, I., The functional response of predator: worries bout scale, TREE, 6, 141-142 (1991)
[10] Holling, C. S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Sec. Can., 45, 1-60 (1965)
[11] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[12] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Rich dynamics of a ratio-dependent one-prey two-predators model, J. Math. Biol., 43, 377-396 (2001) · Zbl 1007.34054
[13] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[14] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[15] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[17] May, R. M., Complexity and Stability in Model Ecosystems (1973), Princeton University Press: Princeton University Press Princeton, NJ
[18] Murry, J. D., Mathematical Biology (1989), Springer: Springer New York
[19] Pang, P. Y.H.; Wang, M. X., Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh, Sect. A, 133, 4, 919-942 (2003) · Zbl 1059.92056
[20] Pang, P. Y.H.; Wang, M. X., Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200, 245-273 (2004) · Zbl 1106.35016
[21] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystem in ecological time, Science, 171, 385-387 (1971)
[22] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 1445-1472 (2001) · Zbl 0986.34045
[23] Takeuchi, Y., Global Dynamical Properties of Lotka-Volterra Systems (1996), World Scientific: World Scientific Singapore · Zbl 0844.34006
[24] Wang, L. L.; Li, W. T., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 167-185 (2003) · Zbl 1029.92025
[25] Wang, L. L.; Li, W. T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response, J. Comput. Appl. Math., 162, 341-357 (2004) · Zbl 1076.34085
[26] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 268-290 (2001) · Zbl 1007.34031
[27] Xiao, D.; Zhang, Z., On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16, 1185-1201 (2003) · Zbl 1042.34060
[28] Zhu, H.; Campebell, S.; Wolkowicz, G., Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63, 636-682 (2002) · Zbl 1036.34049
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