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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:
34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)
34C05Location of integral curves, singular points, limit cycles (ODE)
34C11Qualitative theory of solutions of ODE: growth, boundedness
34C23Bifurcation (ODE)
37G15Bifurcations of limit cycles and periodic orbits
References:
[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 · doi:10.1016/S0025-5564(99)00030-9
[3]Anderson, R.M., May, R.M. (eds.): Population Biology of Infectious Diseases. Springer, Berlin (1982)
[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 · doi:10.1007/s002850050105
[5]Saez, E., Gonzelez-Olivares, E.: Dynamics of a predator–prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999) · Zbl 0934.92027 · doi:10.1137/S0036139997318457
[6]Jost, C., Arino, O., Arditi, R.: About deterministic extinction in ratio-dependent predator–prey models. Bull. Math. Biol. 61, 19–32 (1999) · doi:10.1006/bulm.1998.0072
[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 · doi:10.1016/S0025-5564(01)00049-9
[8]Chattopadhyay, J., Arino, O.: A predator–prey model with disease in the prey. Nonlinear Anal. 36, 747–766 (1999) · Zbl 0922.34036 · doi:10.1016/S0362-546X(98)00126-6
[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 · doi:10.1002/mma.414
[10]Venturino, E.: The influence of disease on Lotka–Volterra systems. Rocky Mountain J. Math. 24, 389–402 (1994)
[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 · doi:10.1016/S0893-9659(03)90096-6
[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 · doi:10.1016/j.chaos.2006.08.010
[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 · doi:10.1016/j.nonrwa.2006.09.004
[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) · doi:10.1090/S0002-9939-1986-0822433-4
[16]Freedman, H.I., Waltman, P.: Persistence in a model of three competitive populations. Math. Biosci. 73(11), 89–101 (1985) · Zbl 0584.92018 · doi:10.1016/0025-5564(85)90078-1
[17]Freedman, H.I., Moson, P.: Persistence definitions and their connections. Proc. Am. Math. Soc. 109, 1025–1033 (1990) · doi:10.1090/S0002-9939-1990-1012928-6
[18]Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–396 (1989) · Zbl 0692.34053 · doi:10.1137/0520025