In this paper, a nonautonomous eco-epidemiological model with disease in the predator is formulated and analyzed, in which saturated predation rate is taken into consideration. Under quite weak assumptions, sufficient conditions for the permanence and extinction of the disease are obtained. Moreover, by constructing a Liapunov function, the global attractivity of the model is discussed. Finally, numerical simulations verified these results.
Krishna, P.D., Chatterjee, S. and Chattopadhyay, J. (2010) Occurrence of Chaos and Its Possible Control in a Predator-Prey Model with Density Dependent Disease-Induced Mortality on Predator Population. Journal of Biological Systems, 18, 399-435. https://doi.org/10.1142/S021833901000339
Chatterjee, S., Bandyopadhyay, M. and Chattopadhyay, J. (2006) Proper Predation Makes the System Disease Free-Conclusion Drawn from an Eco-Epidemiological Model. Journal of Biological Systems, 14, 599-616. https://doi.org/10.1142/S0218339006001970
Xiao, Y. and Chen, L. (2001) Modeling and Analysis of a Predator-Prey Model with Disease in the Prey. Mathematical Biosciences, 171, 59-82. https://doi.org/10.1016/S0025-5564(01)00049-9
Xiao, Y. and Chen, L. (2001) Analysis of a Three Species Eco-Epidemiological Model. Journal of Mathematical Analysis & Applications, 258, 733-754. https://doi.org/10.1006/jmaa.2001.7514
Agarwal, M. and Pandey, P. (2008) A Stage-Structured Predator-Prey Model with Disease in the Prey. Journal of Physics: Conference Series, 96, Article ID: 012118. https://doi.org/10.1088/1742-6596/96/1/012118
Greenhalgh, D. and Haque, M. (2007) A Predator-Prey Model with Disease in the Prey Species only. Mathematical Methods in the Applied Sciences, 30, 911-929. https://doi.org/10.1002/mma.815
Chattopadhyay, J. and Arino, O. (1999) A Predator-Prey Model with Disease in the Prey. Nonlinear Analysis: Theory, Methods & Applications, 36, 747-766. https://doi.org/10.1016/S0362-546X(98)00126-6
Wang, J. and Qu, X. (2011) Qualitative Analysis for a Ratio-Dependent predatorcprey Model with Disease and Diffusion. Applied Mathematics and Computation, 217, 9933-9947. https://doi.org/10.1016/j.amc.2011.04.030
Wuhaib, S.A. and Abu Hasan, Y. (2012) A Prey Predator Model with Vulnerable Infected Prey. Applied Mathematical Sciences, 6, 5333-5348.
Zou, L., Xiong, Z. and Shu, Z. (2011) The Dynamics of an Eco-Epidemic Model with Distributed Time Delay and Impulsive Control Strategy. Journal of the Franklin Institute, 348, 2332-2349. https://doi.org/10.1016/j.jfranklin.2011.06.023
Shi, X., Cui, J. and Zhou, X. (2011) Stability and Hopf Bifurcation Analysis of an Eco-Epidemic Model with a Stage Structure. Nonlinear Analysis: Theory, Methods & Applications, 74, 1088-1106. https://doi.org/10.1016/j.na.2010.09.038
Mandal, P.S. and Banerjee, M. (2012) Deterministic Chaos vs. Stochastic Fluctuation in an Eco-Epidemic Model. Mathematical Modelling of Natural Phenomena, 7, 99-116. https://doi.org/10.1051/mmnp/20127308
Liu, X. and Wang, C. (2010) Bifurcation of a Predator-Prey Model with Disease in the Prey. Nonlinear Dynamics, 62, 841-850. https://doi.org/10.1007/s11071-010-9766-7
Li, J., Gao, W. and Sun, P. (2010) Analysis of a Prey-Predator Model with Disease in Prey. Applied Mathematics and Computation, 217, 4024-4035. https://doi.org/10.1016/j.amc.2010.10.009
Shao, Y., Li, P. and Tang, G. (2012) Dynamic Analysis of an Impulsive Predator-Prey Model with Disease in Prey and Ivlev-Type Functional Response. Abstract & Applied Analysis, 4, 1-16. https://doi.org/10.1155/2012/750530
Kang, A., Xue, Y. and Jin, Z. (2008) Dynamic Behavior of an Eco-Epidemic System with Impulsive Birth. Journal of Mathematical Analysis and Applications, 345, 783-795. https://doi.org/10.1016/j.jmaa.2008.04.043
Liu, J., Zhang, T. and Lu, J. (2012) An Impulsive Controlled Eco-Epidemic Model with Disease in the Prey. Journal of Applied Mathematics and Computing, 40, 459-475. https://doi.org/10.1007/s12190-012-0573-9
Mukherjee, D. (2010) Hopf Bifurcation in an Eco-Epidemic Model. Applied Mathematics and Computation, 217, 2118-2124. https://doi.org/10.1016/j.amc.2010.07.010
Bhattacharyya, R. and Mukhopadhyay, B. (2010) Analysis of Periodic Solutions in an Ecoepidemiological Model with Saturation Incidence and Latency Delay. Nonlinear Analysis: Hybrid Systems, 4, 176-188. https://doi.org/10.1016/j.nahs.2009.09.007
Xue, Y., Kang, A. and Jin, Z. (2008) The Existence of Positive Periodic Solutions of an Eco-Epidemic Model with Impulsive Birth. International Journal of Biomathematics, 1, 327-337. https://doi.org/10.1142/S1793524508000278
Xiao, Y. and Van Den Bosch, F. (2003) The Dynamics of an Eco-Epidemic Model with Biological Control. Ecological Modelling, 168, 203-214. https://doi.org/10.1016/S0304-3800(03)00197-2
Haque, M. (2010) A Predator-Prey Model with Disease in the Predator Species Only. Nonlinear Analysis: Real World Applications, 11, 2224-2236. https://doi.org/10.1016/j.nonrwa.2009.06.012
Zhang, J., Li, W. and Yan, X. (2008) Hopf Bifurcation and Stability of Periodic Solutions in a Delayed Eco-Epidemiological System. Applied Mathematics and Computation, 198, 865-876. https://doi.org/10.1016/j.amc.2007.09.045
Niu, X., Zhang, T. and Teng, Z. (2011) The Asymptotic Behavior of a Nonautonomous Eco-Epidemic Model with Disease in the Prey. Applied Mathematical Modelling, 35, 457-470. https://doi.org/10.1016/j.apm.2010.07.010
Zhang, Y., Gao, S. and Liu, Y. (2012) Analysis of a Nonautonomous Model for Migratory Birds with Saturation Incidence Rate. Communications in Nonlinear Science and Numerical Simulation, 17, 1659-1672. https://doi.org/10.1016/j.cnsns.2011.08.040
Haque, M., Sarwardi, S., Preston, S. and Venturino, E. (2011) Effect of Delay in a Lotka-Volterra Type Predator-Prey Model with a Transmissible Disease in the Predator Species. Mathematical Biosciences, 234, 47-57. https://doi.org/10.1016/j.mbs.2011.06.009
Teng, Z. and Li, Z. (2000) Permanence and Asymptotic Behavior of the N-Species Nonautonomous Lotka Volterra Competitive Systems. Computers & Mathematics with Applications, 39, 107-116. https://doi.org/10.1016/S0898-1221(00)00069-9
Zhang, T. and Teng, Z. (2007) On a Nonautonomous SEIRS Model in Epidemiology. Bulletin of Mathematical Biology, 69, 2537-2559. https://doi.org/10.1007/s11538-007-9231-z
Feng, X., Teng, Z. and Zhang, L. (2008) The Permanence for Nonautonomous N-Species Lotka-Volterra Competitive Systems with Feedback Controls. Rocky Mountain Journal of Mathematics, 38, 1355-1376. https://doi.org/10.1216/RMJ-2008-38-5-1355