The role of transmissible diseases in the Holling-Tanner predator-prey model

Theoretical Population Biology

Volumn 70, Issue 3, 2006, Pages 273-288

  Haque, Mainul ^a   Venturino, Ezio ^b  

^a Krishnath College ^*  (India)

^b UNIVERSITY OF TURIN   (Italy)

Author keywords

Bifurcations; Boundedness; Eco epidemiology; Global stability; Local stability; Persistence

Indexed keywords

BIFURCATION; DISEASE TRANSMISSION; PREDATOR-PREY INTERACTION;

ANIMAL; ANIMAL DISEASE; ARTICLE; BIOLOGICAL MODEL; COMMUNICABLE DISEASE; DEMOGRAPHY; DISEASE TRANSMISSION; ECOLOGY; ENDEMIC DISEASE; EPIDEMIOLOGY; FACTORIAL ANALYSIS; FOOD CHAIN; HUMAN; INCIDENCE; MORTALITY; NONLINEAR SYSTEM; POPULATION DENSITY; POPULATION DYNAMICS; PREDATION; STATISTICAL ANALYSIS; STATISTICAL MODEL; STATISTICS; VALIDATION STUDY;

ANIMALS; COMMUNICABLE DISEASES; DATA INTERPRETATION, STATISTICAL; DEMOGRAPHY; ECOLOGY; ENDEMIC DISEASES; EPIDEMIOLOGY; FACTOR ANALYSIS, STATISTICAL; FOOD CHAIN; HUMANS; INCIDENCE; MODELS, BIOLOGICAL; MODELS, STATISTICAL; MORTALITY; NONLINEAR DYNAMICS; POPULATION DENSITY; POPULATION DYNAMICS; PREDATORY BEHAVIOR;

EID: 33749479525     PISSN: 00405809     EISSN: 10960325     Source Type: Journal    
DOI: 10.1016/j.tpb.2006.06.007     Document Type: Article

References (49)

1. Anderson R.M., and May R.M. Regulation and stability of host-parasite population interactions. I. Regulatory processes. J. Anim. Ecol. 47 (1978) 219-247
2. Anderson R.M., and May R.M. The population dynamics of microparasites and their invertebrates hosts. Proc. R. Soc. London B 291 (1981) 451-463
3. Beltrami E., and Carroll T.O. Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol. 32 (1994) 857-863
4. Berthier K., Langlais M., Auger P., and Pontier D. Dynamics of a feline virus with two transmission models with exponentially growing host populations. Proc. R. Soc. London B 267 (2000) 2049-2056
5. Braza P.A. The bifurcations structure for the Holling Tanner model for predator-prey interactions using two-timing. SIAM. J. Appl. Math. 63 (2003) 889-904
6. Combes C. Les associations du vivant (2001), Flammarion, Paris
7. Combes C. Parasitism, the Ecology and Evolution of Intimate Interactions (2001), University of Chicago Press, Chicago
8. Dawkins R. The Extended Phenotype: The Long Reach of the Gene (1982), Oxford University Press, Oxford
9. De Jong M.C.M., Diekmann O., and Heesterbeek J.A.P. How does infection depend on the population size?. In: Mollison D. (Ed). Epidemic Models, Their Structure and Relation in Data (1994), Cambridge University Press, Cambridge, UK
10. Delgado M., Molina-Becerra M., and Suarez A. Relating disease and predation: equilibria of an epidemic model. Math. Methods Appl. Sci. 28 (2005) 349-362
11. 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28 (1990) 365-382
12. Dobson A., and Crawley M. Pathogens and the structure of plant communities. Trends Ecol. Evol. 9 (1994) 393-398
13. Freedman H.I. Deterministic Mathematical Method in Population Ecology (1980), Marcel Dekker, New York
14. Freedman H.I. A model of predator-prey dynamics modified by the action of parasite. Math. Biosci. 99 (1990) 143-155
15. Gao L.Q., and Hethcote H.W. Disease transmission models with density-dependent demographics. J. Math. Biol. 30 (1992) 717-731
16. Gibson C.C., and Watkinson A.R. The role of the hemi parasite annual Rhinanthus minor in determining grass land community structure. Oecologia 89 (1992) 62-68
17. Hadeler K.P., and Freedman H.I. Predator-prey populations with parasitic infection. J. Math. Biol. 27 (1989) 609-631
18. Hale J. Ordinary Differential Equations (1989), Krieger Publisher Co., Malabar
19. Hamilton W.D., Axelrod R., and Tanese R. Sexual reproduction as an adaptation to resist parasite. Proc. Nat. Acad. Sci. (USA) 87 (1990) 3566-3573
20. Hammond A.M., and Hardy T.N. Quality of diseased plants as host for insects. In: Heinrichs E.A. (Ed). Plant Stress-insect Interactions (1989), Wiley, New York 381-432
21. Han L., Ma Z., and Hethcote H.W. Four predator prey models with infectious diseases. Math. Comp. Model. 30 (2001) 849-858
22. Haque, M., Venturino, E., 2006. Increase of the prey may decrease the healthy predator population in presence of a disease in the predator Hermis (to appear).
23. Hechtel L.J., John C., and Huliano S.A. Modification of antipredator behavior of caecidotea intermedius by its parasite Acanthocephalus dirus. Ecology 74 (1993) 710-713
24. Hethcote H.W. The mathematics of infectious diseases. SIAM Rev. 42 (2000) 599-653
25. Hethcote H.W., Ma Z., and Liao S. Effects of quarantine in six endemic models for infectious diseases. Math. Biosci. 180 (2002) 141-160
26. Hethcote H.W., Wang W., and Ma Z. A predator prey model with infected prey. Theoret. Popul. Biol. 66 (2004) 259-268
27. Holling C.S. The functional response of invertebrate predators to prey density. Memories Entomol. Soc. Can. 45 (1965) 3-60
28. Hsu S.B., and Hwang T.W. Global stability for a class of predator prey system. Siam. J. Appl. Math. 55 (1995) 763-783
29. Hudson P.J., Dobson A.P., and Newborn D. Do the parasites make prey vulnerable to predation? Red grouse and parasites. J. Anim. Ecol. 61 (1992) 681-692
30. Kermack W.O., and Mc Kendrick A.G. Contributions to the mathematical theory of epidemics, part 1. Proc. R. Soc. London Ser. A 115 (1927) 700-721
31. Linhart Y.B. Disease, Parasitism and Herbivore multidimensional challenges in plant evolution. Trends Ecol. Evol. 6 (1991) 392-396
32. Manfredi P., and Salinelli E. Population-induced oscillations in blended SI-SEI epidemiological models. IMA J. Math. Appl. Med. Biol. 19 (2002) 95-112
33. May R.M. Limit cycles in predator-prey communities. Science 177 (1972) 900-902
34. May R.M. Stability and complexity in Model Ecosystem (1974), Princeton University Press, Princeton, NJ
35. Mena-Lorca J., and Hethcote H.W. Dynamic models of infectious diseases as regulator of population sizes. J. Math. Biol. 30 (1992) 693-716
36. Minchella D.J., and Scott M.E. Parasitism a tropic determinant of animal community structure. Trends Ecol. Evol. 6 (1991) 250-254
37. Murray J.D. Mathematical Biology (1989), Springer, New York
38. Renshaw E. Modelling Biological Populations in Space and Time (1991), Cambridge University Press, Cambridge, UK
39. Saez E., and Gonzelez-Olivares E. Dynamics of a predator-prey model. SIAM J. Appl. Math. 59 (1999) 1867-1878
40. Singh B.K., Chattopadhyay J., and Sinha S. The role of virus infection in a simple phytoplankton zooplankton system. J. Theo. Biol. 231 (2004) 153-166
41. Sotomayor J. Generic bifurcations of dynamical systems. In: Peixoto M.M. (Ed). Dynamical Systems (1973), Academic Press, New York 549-560
42. Tanner J.T. The stability and intrinsic growth rates of prey and predator populations. Ecology 56 (1975) 855-867
43. Temple S.A. Do predator always capture subtended individuals disproportionately from prey population?. Ecology 68 (1987) 669-674
44. Venturino E. The influence of diseases on Lotka-Volterra systems. Rocky Mount. J. Math. 24 (1994) 381-402
45. Venturino, E., 1995. Epidemics in predator-prey models: disease in prey. In: O. Arino, D. Axelrod, M., Kimmel, Langlais, M. (Eds.), Mathematical Population dynamics, Analysis of Heterogeneity, vol. 1, pp. 381-393.
46. Venturino E. The effects of diseases on competing species. Math. Biosc. 174 (2001) 111-131
47. Venturino E. Epidemics in predator-prey model: disease in the predators. IMA J. Math. Appl. Med. Biol. 19 (2002) 185-205
48. Wollkind D.J., Collings J.B., and Logan J.A. Metastability in a temperature-dependent model system for predator prey mite outbreak interactions on fruit flies. Bull. Math. Biol. 50 (1988) 379-409
49. Xiao Y., and Chen L. Modelling and analysis of a predator-prey model with disease in prey. Math. Biosci. 171 (2001) 59-82