Stability and Hopf bifurcation of a delay eco‐epidemiological model with nonlinear incidence rateFootnote*

This work is supported by the National Natural Science Foundation of China (No. 10771104 and No.10471117), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 2010IRTSTHN006) and Program for Key Laboratory of Simulation and Control for Population Ecology in Xinyang Normal University (No. 201004) and Natural Science Foundation of the Education Department of Henan Province (No. 2009B1100200 and No. 2010A110017)

References

• Aziz‐Alaoui , M.A. 2002 . Study of a Leslie‐Gower‐type tritrophic population . Chaos Solitons & Fractals , 14 : 1275 – 1293 . doi:10.1016/S0960–0779 (02)00079–6.
   Web of Science ®Google Scholar
• Aziz‐Alaoui , M.A. and Okiye , M. Daher . 2003 . Boundedness and global stability for a predator‐prey model with modified Leslie‐Gower and Holling‐type II schemes . Appl. Math. Lett. , 16 : 1069 – 1075 .
   Web of Science ®Google Scholar
• Beretta , E. and Kuang , Y. 2002 . Geometric stability switch criteria in delay differential systems with delay dependent parameters . SIAM J. Math. Anal. , 33 : 1144 – 1165 . doi:10.1137/S0036141000376086.
   Web of Science ®Google Scholar
• Bhunu , C.P. and Garira , W. 2009 . A two strain tuberculosis transmission model with therepy and quarantine . Math. Model. Anal. , 14 (3) : 291 – 312 . doi:10.3846/1392–6292.2009.14.291–312.
   Web of Science ®Google Scholar
• Butler , G. , Freedman , H.I. and Waltman , P. 1986 . Uniformly persistence system . Proc. Am. Math. Soc. , 96 : 425 – 430 . doi:10.1090/S0002–9939–1986–0822433–4.
   Web of Science ®Google Scholar
• Capasso , V. and Serio , G. 1978 . A generalization of the Kermack‐Mckendrick deterministic epidemic model . Math Biosci. , 42 : 41 – 61 . doi:10.1016/0025–5564 (78)90006–8.
   Web of Science ®Google Scholar
• Freedman , H.I. and Moson , P. 1990 . Persistence definitions and their connections . Proc. Am. Math. Soc. , 109 : 1025 – 1033 . doi:10.1090/S0002–9939–1990–1012928–6.
   Web of Science ®Google Scholar
• Freedman , H.I. and Rao , V.S.H. 1983 . The trade‐off between mutual interference and time lags in predator‐prey systems . Bulletin of Mathematical Biology , 45 : 991 – 1004 . doi:10.1007/BF02458826.
   Web of Science ®Google Scholar
• Freedman , H.I. and Waltman , P. 1985 . Persistence in a model of three competitive populations . Mathematical Biosciences , 73 (11) : 89 – 101 . doi:10.1016/0025–5564 (85)90078–1.
   Google Scholar
• Gopalsamy , K. 1992 . Stability and oscillation in delay different equations of population dynamics , Boston : Kluwer Academic Publishers .
   Google Scholar
• Greenhalgh , D. and Haque , M. 2007 . A predator‐prey model with disease in the prey species only . Mathematical Methods in the Applied Sciences , 30 (8) : 911 – 929 . doi:10.1002/mma.815.
   Web of Science ®Google Scholar
• Hale , J.K. and Waltman , P. 1989 . Persistence in infinite‐dimensional systems . SIAM J. Math. Anal. , 20 : 388 – 396 . doi:10.1137/0520025.
   Web of Science ®Google Scholar
• Haque , M. and Greenhalgh , D. 2010 . When a predator avoids infected prey: a model‐based theoretical study . Math. Med. Biol. , 27 (1) : 75 – 94 . doi:10.1093/imammb/dqp007.
   PubMed Web of Science ®Google Scholar
• Haque , M. and Venturino , E. 2006 . The role of transmissible disease in Holling‐Tanner predator‐prey model . Theoretical Population Biology , 70 : 273 – 288 . doi:10.1016/j.tpb.2006.06.007.
   PubMed Web of Science ®Google Scholar
• Haque , M. and Venturino , E. 2009 . “ Modelling disease spreading in symbiotic communities? ” . In Wildlife: Destruction, Conservation and Biodiversity , Edited by: Harris , John D. and Brown , Paul L. 135 – 180 . New York : Nova Science Publishers .
   Google Scholar
• Hassard , B.D. , Kazarinoff , N.D. and Wan , Y.H. 1981 . Theory and application of Hopf bifucation , Cambridge : Cambridge University .
   Google Scholar
• Hethcote , H.W. 2000 . The mathematics of infectious disease . SIAM Review , 42 : 599 – 653 . doi:10.1137/S0036144500371907.
   Web of Science ®Google Scholar
• Hethcote , H.W. , Wang , W. , Han , L. and Zhien , Ma . 2004 . A predator‐prey model with infected prey . Theor. Pop. Biol. , 66 : 259 – 268 . doi:10.1016/j.tpb.2004.06.010.
   PubMed Web of Science ®Google Scholar
• Hsu , S.B. and Hwang , T.W. 1999 . Hopf bifurcation analysis for a predator‐prey system of Holling and Leslie type . Taiwan J. Math. , 3 : 35 – 53 .
   Web of Science ®Google Scholar
• Hsu , S.B. , Hwang , T.W. and Kuang , Y. 2001 . Global analysis of the Michaelis‐Menten type ratio‐dependent predator‐prey system . J. Math. Biol. , 42 : 489 – 506 . doi:10.1007/s002850100079.
   PubMed Web of Science ®Google Scholar
• Hudson , P.J. , Dobson , A.P. and Newborn , D. 1992 . Do parasites make prey vulnerable to predation? Red grouse and parasites . J. Anim. Ecol. , 61 : 681 – 692 . doi:10.2307/5623.
   Web of Science ®Google Scholar
• Hurwitz , A. 1895 . On the conditions under which an equation has only roots with negative real parts . Math. Ann. , 46 : 273 – 284 .
   Google Scholar
• Hwang , T.W. 1999 . Uniqueness of limit cycle for Gause‐type predator‐prey systems . J. Math. Anal. Appl. , 238 : 179 – 195 . doi:10.1006/jmaa.1999.6520.
   Web of Science ®Google Scholar
• Kuang , Y. and Beretta , E. 1998 . Global qualitative analysis of a ratio‐dependent predator‐prey system . J. Math. Biol. , 36 : 389 – 406 . doi:10.1007/s002850050105.
   Web of Science ®Google Scholar
• Lafferty , K.D. and Morris , A.K. 1996 . Altered behaviour of parasitized killfish increases susceptibility to predation by bird final hosts . Ecology , 77 : 1390 – 1397 . doi:10.2307/2265536.
   Web of Science ®Google Scholar
• Liu , M.X. , Jin , Z. and Haque , M. 2009 . The periodic predator‐prey Lotka‐Volterra model with disease in the prey . Nonlinear Analysis: Real World Applications , 10 : 3098 – 3111 . doi:10.1016/j.nonrwa.2008.10.010.
   Web of Science ®Google Scholar
• Liu , W.M. , Hethcote , H.W. and Levin , S.A. 1987 . Dynamical behavior of epidemiological models with nonlinear incidence rates . J. Math. Biol. , 25 : 359 – 380 . doi:10.1007/BF00277162.
   PubMed Web of Science ®Google Scholar
• Magombedze , G. , Mukandavire , Z. , Chiyaka , C. and Musuka , G. 2009 . Optimal control of a sex‐structured HIV/AIDS model with condom use . Math. Model. Anal. , 14 (4) : 483 – 494 . doi:10.3846/1392–6292.2009.14.483–494.
   Web of Science ®Google Scholar
• Murray , D.L. , Carry , J.R. and Keith , L.B. 1997 . Interactive effects of sublethal nematodes and nutritional status on snowshoe hare vulnerability to predation? . J. Anim. Ecol. , 66 : 250 – 264 . doi:10.2307/6026.
   Web of Science ®Google Scholar
• Nindjin , A.F. , Aziz‐Alaoui , M.A. and Cadivel , M. 2006 . Analysis of a predator‐prey model with modified Leslie‐Gower and Holling‐type II schemes with time delay . Nonlinear Analysis: Real World Applications , 7 (5) : 1104 – 1118 . doi:10.1016/j.nonrwa.2005.10.003.
   Web of Science ®Google Scholar
• Samanta , G.P. 2010 . Analysis of a nonautonomous HIV/AIDS epidemic model with distributed time delay . Math. Model. Anal. , 15 (3) : 327 – 347 . doi:10.3846/1392–6292.2010.15.327–347.
   Web of Science ®Google Scholar
• Song , X.Y. and Li , Y.F. 2008 . Dynamic behaviors of the periodic predatorprey model with modified Leslie‐Gower Holling‐type II schemes and impulsive effect . Nonlinear Analysis: Real World Applications , 9 (1) : 64 – 79 . doi:10.1016/j.nonrwa.2006.09.004.
   Web of Science ®Google Scholar
• Venturino , E. 1995 . “ Epidemics in predator prey models: disease in the prey ” . In Mathematical Population Dynamics, Analysis of Heterogeneity , Theory of Epidemics Edited by: Arino , O. volume 1 , 381 – 393 . Winnipeg, , Canada : Wuertz Publishing Ltd. .
   Google Scholar
• Wang , W. and Ma , Z. 2002 . Global dynamics of an epidemic model with time delay . Nonlinear Anal. RWA , 3 : 365 – 373 . doi:10.1016/S1468–1218 (01)00035–9.
   Google Scholar
• Xiao , Y.N. and Chen , L.S. 2001a . Analysis of a three species eco‐epidemiological model . Journal of Mathematical Analysis and Applications , 258 (2) : 733 – 754 . doi:10.1006/jmaa.2001.7514.
   Web of Science ®Google Scholar
• Xiao , Y.N. and Chen , L.S. 2001b . Modeling and analysis of a predator‐prey model with disease in the prey . Math. Biosci. , 171 : 59 – 82 . doi:10.1016/S0025–5564 (01)00049–9.
   PubMed Web of Science ®Google Scholar
• Xu , R. and Ma , Z.E. 2009a . Global stability of a SIR epidemic model with nonlinear incidence rate and time delay . Nonlinear Analysis: Real World Applications , 10 (5) : 3175 – 3189 . doi:10.1016/j.nonrwa.2008.10.013.
   Web of Science ®Google Scholar
• Xu , R. and Ma , Z.E. 2009b . Stability of a delayed SIRS epidemic model with a nonlinear incidence rate . Chaos, Solitons & Fractals , 41 (5) : 2319 – 2325 . doi:10.1016/j.chaos.2008.09.007.
   Web of Science ®Google Scholar
• Yang , X. , Chen , L.S. and Chen , J.F. 1996 . Permanence and positive periodic solution for the single‐species nonautonomous delay diffusive models . Computers and Mathematics with Applications , 32 : 109 – 116 . doi:10.1016/0898–1221 (96)00129–0.
   Web of Science ®Google Scholar
• This work is supported by the National Natural Science Foundation of China (No. 10771104 and No.10471117), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 2010IRTSTHN006) and Program for Key Laboratory of Simulation and Control for Population Ecology in Xinyang Normal University (No. 201004) and Natural Science Foundation of the Education Department of Henan Province (No. 2009B1100200 and No. 2010A110017)
   Google Scholar