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Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system. (English) Zbl 1134.92034
Summary: A delayed predator-prey epidemiological system with disease spreading in the predator population is considered. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcations of periodic solutions are investigated. Moreover, we also study the direction of Hopf bifurcations and the stability of bifurcated periodic solutions; an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

MSC:
92D30 Epidemiology
92D40 Ecology
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
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[1] Chattopadhyay, J.; Arino, O., A predator – prey model with disease in the prey, Nonlinear anal. TMA, 36, 747-766, (1999) · Zbl 0922.34036
[2] Fan, M.; Li, M.; Wang, K., Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. biosci., 170, 199-208, (2001) · Zbl 1005.92030
[3] Gao, S.; Chen, L.; Sun, L., Dynamic complexities in a seasonal prevention epidemic model with birth pulses, Chaos soliton fract., 26, 1171-1181, (2005) · Zbl 1064.92039
[4] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.H., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
[5] Huo, H.F.; Li, W.T., Periodic solution of a delayed predator – prey system with michaelis – menten type functional response, J. comput. appl. math., 166, 453-463, (2004) · Zbl 1047.34081
[6] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[7] Kermack, W.O.; McKendrick, A.G., A contribution to the mathematical theory of epidemics, Proc. roy. soc. lond. A, 115, 700-721, (1927) · JFM 53.0517.01
[8] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[9] Li, M.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. biosci., 160, 191-213, (1999) · Zbl 0974.92029
[10] Li, W.T.; Wang, L.L., Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control, J. comput. appl. math., 180, 293-309, (2005) · Zbl 1069.34100
[11] Mukherjee, D., Stability analysis of a stochastic model for prey – predator system with disease in the prey, Nonlinear anal.: modell. contr., 8, 83-92, (2003) · Zbl 1042.92039
[12] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. diff. eq., 188, 135-163, (2003) · Zbl 1028.34046
[13] Sun, C.; Lin, Y.; Han, M., Stability and Hopf bifurcation for an epidemic disease model with delay, Chaos soliton fract., 30, 204-216, (2006) · Zbl 1165.34048
[14] Sun, S.; Yuan, C., On the analysis of predator – prey model with epidemic in the predator, J. biomath., 21, 97-104, (2006) · Zbl 1119.92061
[15] Song, Y.; Han, M.; Wei, J., Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D, 200, 185-204, (2005) · Zbl 1062.34079
[16] Wang, L.L.; Fan, Y.H.; Li, W.T., Multiple bifurcations in a predator – prey system with monotonic functional response, Appl. math. comput., 172, 1103-1120, (2006) · Zbl 1102.34031
[17] Xiao, Y.; Chen, L., Analysis of a three species eco-epidemiological model, J. math. anal. appl., 258, 733-754, (2001) · Zbl 0967.92017
[18] Yu, W.; Cao, J., Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay, Nonlinear anal. TMA, 62, 141-165, (2005) · Zbl 1138.34348
[19] Yan, X.P.; Li, W.T., Bifurcation and global periodic solutions in a delayed facultative mutualism system, Physica D, 227, 51-69, (2007) · Zbl 1123.34055
[20] Yan, X.P.; Li, W.T., Hopf bifurcation and global periodic solutions in a delayed predator – prey system, Appl. math. comput., 177, 427-445, (2006) · Zbl 1090.92052
[21] Zhang, J.; Ma, Z., Global dynamics of an SEIR epidemic model with saturating contact rate, Math. biosci., 185, 15-32, (2003) · Zbl 1021.92040
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