Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. (English) Zbl 1063.34044

The authors present a two-dimensional differential system modeling a predator-prey food chain, and based on a modified version of the Leslie-Gower scheme and on the Holling-type II scheme. The main result is given in terms of boundedness of solutions, existence of an attracting set and global stability of the coexisting interior equilibrium.


34D23 Global stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] Letellier, C.; Aziz-Alaoui, M. A., Analysis of the dynamics of a realistic ecological model, Chaos Solitons and Fractals, 13, 1, 95-107 (2002) · Zbl 0977.92029
[2] Letellier, C.; Aguirré, L.; Maquet, J.; Aziz-Alaoui, M. A., Should all the species of a food chain be counted to investigate the global dynamics, Chaos Solitons and Fractals, 13, 5, 1099-1113 (2002) · Zbl 1004.92039
[3] Upadhyay, R. K.; Rai, V., Why chaos is rarely observed in natural populations, Chaos Solitons and Fractals, 8, 12, 1933-1939 (1977)
[4] Aziz-Alaoui, M. A., Study of a Leslie-Gower-type tritrophic population, Chaos Sol. and Fractals, 14, 8, 1275-1293 (2002) · Zbl 1031.92027
[5] Korobeinikov, A., A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14, 6, 697-699 (2001) · Zbl 0999.92036
[6] Hanski, I. L.; Hansson, L.; Henttonen, H., Specialist predators, generalist predators and the microtine rodent cycle, J. Animal Ecology, 60, 353-367 (1991)
[7] Leslie, P. H., Some further notes on the use of matrices in population mathematics, Biometrica, 35, 213-245 (1948) · Zbl 0034.23303
[8] Leslie, P. H.; Gower, J. C., The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrica, 47, 219-234 (1960) · Zbl 0103.12502
[9] Pielou, E. C., An Introduction to Mathematical Ecology (1969), Wiley-Interscience: Wiley-Interscience New York · Zbl 0259.92001
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