×

zbMATH — the first resource for mathematics

Modelling a predator-prey system with infected prey in polluted environment. (English) Zbl 1193.34071
Summary: A predator-prey model with logistic growth in prey is modified by introducing an SIS parasite infection in the prey. We have studied the combined effect of environmental toxicant and disease on prey-predator system. It is assumed in this paper that the environmental toxicant affects both prey and predator population and the infected prey is assumed to be more vulnerable to the toxicant and predation compared to the sound prey individuals. Thresholds are identified which determine when system persists and disease remains endemic.

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
92D25 Population dynamics (general)
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zhien, Ma.; Jun, Song Bao; Hallam, T.G., The threshold of survival for system in a fluctuating environment, Bull. math. biol., 51, 3, 311-323, (1989) · Zbl 0676.92010
[2] Ma, Zhien; Hallam, T.G., Effects of parameter fluctuations on community survival, Math. biosci., 86, 35-49, (1987) · Zbl 0631.92019
[3] Liu, Huaping; Ma, Zhien, The threshold of survival for system of two species in a polluted environment, J. math. biol., 30, 49-61, (1990) · Zbl 0745.92028
[4] Li, Zhan; Zhisheng, Shun; Wang, Ke, Persistence and extinction of single population in a polluted environment, Elect. J. diff. eqs., 108, 1-5, (2004) · Zbl 1084.92032
[5] Curtis, H., Invitation to biology, (1972), Worth Publisher New York
[6] Sih, A.; Crowly, P.; McPeek, M.; Petranka, J.; Strohmeier, K., Predation, competition and prey communities:a review of field experiments, Ann. rev. ecol. semant., 16, 269-311, (1985)
[7] Holmes, J.C.; Bethel, W.M., Modifications of intermediate host behaviour by parasites, (), 123-149
[8] Dobson, A.P., The population of biology of parasite-induced changes in host behaviour, Quart. rev. biol., 63, 139-165, (1988)
[9] Moore, J., Parasites and the behaviour of animals, (2002), Oxford University Press Oxford
[10] Chattopadhyay, J.; Arino, O., A predator – prey model with disease in prey, Nonlinear anal., 36, 747-766, (1999) · Zbl 0922.34036
[11] Arkoosh, M.R.; Casillas, E.; Clemons, E.; Kagley, Anna N.; Olson, R.; Reno, P.; Stein, John., Effect of pollution on fish diseases: potential impacts on salmonid populations, J. aquat. anim. health, 10, 182-190, (1998)
[12] Hethcote, H.W., Qualitative analysis of communicable disease models, Math. biosci., 28, 335-356, (1976) · Zbl 0326.92017
[13] ()
[14] De Jong, M.C.M.; Diekmann, O.; Heesterbeek, J.A.P., How does transmission depend on population size?, (), 84-94 · Zbl 0850.92042
[15] Hethcote, H.W.; Ma, Z.; Liao, Shengbing, Effects of quaratine in six epidemic models for infectious diseases, Math. biosci., 180, 141-160, (2002) · Zbl 1019.92030
[16] Hethcote, H.W.; Wang, W.; Han, Litao; Ma, Zhien, A predator – prey model with infected prey, Theor. popul. biol., 66, 259-268, (2004)
[17] Thieme, H.R., Convergence results and a poincare – bendixon trichotomy for asymptotically autonomous differential equations, J. math. biol., 30, 755-763, (1992) · Zbl 0761.34039
[18] Han, L.; Ma, Z.; Hethcote, H.W., Four predator – prey models with infectious diseases, Math. comp. modell., 34, 849-858, (2001) · Zbl 0999.92032
[19] Hutson, V., A theorem on average Liapunov function, Monatsh. math., 98, 267, (1984) · Zbl 0542.34043
[20] Thieme, H.R., Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. math. anal., 24, 407-435, (1993) · Zbl 0774.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.