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A predator-prey model with infection in both populations
|關鍵字:||predator-prey model;捕食模式;disease infection;global stability analysis;basic reproduction numbers;unique periodic solution;傳染病;穩定性分析;分歧現象;傳染基數;週期解||出版社:||應用數學系所||引用:|| W. H. Van Dobben, The food of cormorants in the Netherlands, Ardea, 40: 1-63 (1952)  G. E. Vaughn and P. W. Coble, Sublethal effects of three ectoparasites on ﬁsh, J. Fish-eries Biol., 7: 283-294 (1975)  S. A. Temple, Do predators always capture substandard individuals disproportionately from prey populations?, Ecology, 68: 669-674 (1987)  J. C. Holmes and W. M. Bethel, Modiﬁcations of intermediate host behaviour by para-sites. In: E. V. Canning, C. A. Wright (Eds.), Behavioural Aspects of Parasite Trans-mission, Suppl I to Zool. f. Linnean Soc., 51: 123-149 (1972)  A. P. Dobson, The population biology of parasite-induced changes in host behavior, Q. Rev. Biol., 63: 139-165 (1988)  J. Moore, Parasites and the Behaviour of Animals, Oxford University Press, Oxford (2002)  J. R. Krebs, Optimal foraging: decision rules for predators. In: J. R. Krebs, N. B. Da-vies (Eds.), Behavioural Ecology: an Evolutionary approach, First ed. Blackwell Scientiﬁc Publishers, Oxford, 23-63 (1978)  R. O. Peterson and R. E. Page, The rise and fall of Isle Royale wolves, 1975-1986. J. Mamm., 69 (I): 89-99 (1988)  L. D. Mech, The Wolf, Natural History Press, New York (1970)  G. B. Schaller, The Serengeti Lion: A Study of Predator Prey Relations, University of Chicago Press, Chicago (1972)  R. M. Anderson and R. M. May, The invasion, persistence, and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. Lond., B 314: 533-570 (1986)  K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27: 609-631 (1989)  E. Venturino, The influence of diseases on Lotka-Volterra systems. Rky. Mt. J. Math., 24: 381-402 (1994)  E. Venturino, Epidemics in predator-prey models: Disease in the prey. In: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analy-sis of Heterogeneitiy, vol. 1:Theory of Epidemics, Wuerz Publishing, Winnipeg, Canada, 381-393 (1995)  J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36: 747-766 (1999)  L. Han, Z. Ma and H. W. Hethcote, Four Predator Prey Models with Infectious Disease, Mathematical and Computer Modelling, 34: 849-858 (2001)  H. W. Hethcote, W. Wang, L. Han and Z. Ma, A Predator-prey Model with Infected Prey, Theoretical Population Biology, 66: 259-268 (2004)  H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, Inc., New York (1980)  D. Butler, Bird-flu experts question advice on eating poultry, Nature 440, 850-851 (2006)  S. B. Hsu and Y. H. Hsieh, Modeling Intervention Measures and Severity-Dependent Public Response during Severe Acute Respiratory Syndrome Outbreak, SIAM J. Appl. Math., 66. 627-647 (2006)  F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, Inc., New York (1969)  L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equa-tions, 2nd edition, Springer, Berlin (1963)  E. A. Coddington and N. Levinson, Theory of Differential Equations, McGraw-Hill, New York (1955)  S. B. Hsu, S. P. Hubbell and P. Waltman, Competing Predators, SIAM J. Appl. Math., 35: 621-622 (1978)  W. Hahn, Stability of Motion, Springer, New York (1967)  K. S. Cheng, Uniqueness of A Limit Cycle for A Predator-Prey System, SIAM J. Math. Anal., 12. 541-548 (1981)  Y. Kuang and H.I. Freedman, Uniqueness of Limit Cycle in Gause-Type Models of Predator-Prey Systems, Math. Biosciences, 88: 67-84 (1988)  E. C. Pielou, Introduction to Mathematical Ecology, Wiley-Interscience, New York (1969)||摘要:||
In this work, we propose a predator-prey model with disease infection in both populations. Global analysis is given for the disease free model. For the full model with disease, we obtain five threshold parameters (or basic reproduction numbers). Coupled conditions given in the form of the threshold parameters are needed for the local stability of the equilibria under consideration. One of the conditions, in the form of a basic reproduction number for the predator-prey system, always determines the coexistence of the predators and prey; the other condition, in the form of a basic reproduction number for the infectious disease, dictates whether the disease will be eradicated. This phenomenon of dual conditions was also ob-served by Hethcote et al. [16, 17] in their studies of predator-prey model with infection in the prey population. In some cases, the existence of a unique periodic solution which is orbitally stable is obtained through numerical simulation. The mathematical proof of uniqueness and existence of periodic solution, as well as the stability of endemic positive equilibrium, are still the open problems.
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