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A modified Leslie-Gower predator-prey model with prey infection. (English) Zbl 1205.34061
The authors study a predator-prey model where the prey population is divided into two groups with susceptible and infected individuals, whereas predation affects only the infected prey through a modified Holling II functional response. A boundedness result is proved and the local stability of all nonnegative equilibria is analyzed. Under certain conditions, the system undergoes Andronov-Hopf bifurcation at the interior positive equilibrium. The results of numerical simulation are presented.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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[1] Anderson, R.M., May, R.M.: Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, Oxford (1991)
[2] Li, M.Y., Graef, J.R., Wang, L.C., Karsai, J.: Global dynamics of a SEIR model with a varying total population size. Math. Biosci. 160, 191–213 (1999) · Zbl 0974.92029
[3] Anderson, R.M., May, R.M. (eds.): Population Biology of Infectious Diseases. Springer, Berlin (1982) · Zbl 1225.37099
[4] Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio dependent predator prey system. J. Math. Biol. 36, 389–406 (1998) · Zbl 0895.92032
[5] Saez, E., Gonzelez-Olivares, E.: Dynamics of a predator–prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999) · Zbl 0934.92027
[6] 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
[7] Xiao, Y.N., Chen, L.S.: Modelling and analysis of a predator–prey model with disease in the prey. Math. Biosci. 171, 59–82 (2001) · Zbl 0978.92031
[8] Chattopadhyay, J., Arino, O.: A predator–prey model with disease in the prey. Nonlinear Anal. 36, 747–766 (1999) · Zbl 0922.34036
[9] Chattopadhyay, J., Pal, S., El Abdllaoui, A.: Classical predator–prey system with infection of prey population – a mathematical model. Math. Methods Appl. Sci. 26, 1211–1222 (2003) · Zbl 1044.34001
[10] Venturino, E.: The influence of disease on Lotka–Volterra systems. Rocky Mountain J. Math. 24, 389–402 (1994) · Zbl 0799.92017
[11] Aziz-alaoui, M.A., Daher Okiye, M.: Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II type schemes. Appl. Math. Lett. 16, 1069–1075 (2003) · Zbl 1063.34044
[12] Guo, H.J., Song, X.Y.: An impulsive predator–prey system with modified Leslie–Gower and Holling type II schemes. Chaos, Solitons Fractals 36(5), 1320–1331 (2008) · Zbl 1148.34034
[13] Song, X.Y., Li, Y.F.: Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect. Nonlinear Anal. Real World Appl. 9(1), 64–79 (2008) · Zbl 1142.34031
[14] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, London (2004)
[15] Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistence system. J. Proc. Am. Math. Soc. 96, 425–430 (1986) · Zbl 0603.34043
[16] Freedman, H.I., Waltman, P.: Persistence in a model of three competitive populations. Math. Biosci. 73(11), 89–101 (1985) · Zbl 0584.92018
[17] Freedman, H.I., Moson, P.: Persistence definitions and their connections. Proc. Am. Math. Soc. 109, 1025–1033 (1990) · Zbl 0695.34049
[18] Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–396 (1989) · Zbl 0692.34053
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