zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Predator-prey populations with parasitic infection. (English) Zbl 0716.92021
Summary: A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.
MSC:
92D30Epidemiology
34C05Location of integral curves, singular points, limit cycles (ODE)
92D40Ecology
References:
[1]Anderson, R. M., May, R. M.: The invasion, persistence, and spread of infectious diseases within animal and plant communities. Philos. Trans. R. Soc. Lond., B 314, 533–570 (1986) · doi:10.1098/rstb.1986.0072
[2]Aron, J. L., May, R. M.: The population dynamics of malaria. In: Anderson, R. M. (ed.) The population dynamics of infectious diseases, theory and application, pp. 139–179. London: Chapman and Hall 1982
[3]Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H. P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-parameter semigroups of positive operators. (Lect. Notes Math., vol. 1184) Berlin Heidelberg New York: Springer 1986
[4]Butler, G. J., Freedman, H. I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96, 425–430 (1986) · doi:10.1090/S0002-9939-1986-0822433-4
[5]Curio, E.: Behavior and parasitism, chap. 2. In: Mehlhom, K. (ed.) Parasitology in focus, pp. 149–160. Berlin Heidelberg New York: Springer 1988
[6]Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. (Lect. Notes Biomath., vol. 11, pp. 1–14) Berlin Heidelberg New York: Springer 1976
[7]Dobson, A. P.: The population biology of parasite-induced changes in host behavior. Q. Rev. Biol. 63, 139–165 (1988) · doi:10.1086/415837
[8]Freedman, H. I.: Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci. 31, 207–225 (1976) · Zbl 0373.92023 · doi:10.1016/0025-5564(76)90080-8
[9]Freedman, H. I.: Deterministic mathematical models in population ecology. HFR Consulting Ltd.: Edmonton, 1987
[10]Freedman, H. I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984) · Zbl 0534.92026 · doi:10.1016/0025-5564(84)90032-4
[11]Freedman, H. I., Waltman, P.: Persistence in a model of three competitive populations. Math. Biosci. 73, 89–101 (1985) · Zbl 0584.92018 · doi:10.1016/0025-5564(85)90078-1
[12]Freedman, H. I., Wolkowicz, G. S. K.: Predator-prey systems with group defense: the paradox of enrichment revisited. Bull. Math. Biol. 48, 493–508 (1986)
[13]Gantmacher, F. R.: The theory of matrices, Chap. 13. Chelsea 1959
[14]Hadeler, K. P., Dietz, K.: Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations. Comput. Math. Appl. 9, 415–430 (1983) · Zbl 0518.92021 · doi:10.1016/0898-1221(83)90056-1
[15]Hadeler, K. P.: Spread and age structure in epidemic models. In: Perspectives in Mathematics, Anniversary of Oberwolfach. Basel: Birkhäuser 1984
[16]Hadeler, K. P., Dietz, K.: Population dynamics of killing parasites which reproduce in the host. J. Math. Biol. 21, 45–65 (1984) · Zbl 0554.92015 · doi:10.1007/BF00275222
[17]Hofbauer, J., Sigmund, K.: Permanence for replicator equations. In: Kurzhansky, A. B., Sigmund, K.: Dynamical systems. (Lect. Notes. Econ. Math. Syst., vol. 287) Berlin Heidelberg New York: Springer 1987
[18]Holling, C. S.: Some characteristics of simple types of predation and parasitism, Can. Ent. 91, 385–398 (1959) · doi:10.4039/Ent91385-7
[19]Holmes, J. C., Bethel, W. M.: Modification of intermediate host behaviour by parasites. In: Canning, E. V., Wright, C. A. (eds.) Behavioural aspects of parasite transmission. Suppl. I to Zool. f. Linnean Soc. 51, 123–149 (1972)
[20]Kermack, W. O., McKendrick, A. G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond., A 115, 700–721 (1927) · Zbl 02582630 · doi:10.1098/rspa.1927.0118
[21]Kuang, Y., Freedman, H. I.: Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biosci. 88, 67–84 (1988) · Zbl 0642.92016 · doi:10.1016/0025-5564(88)90049-1
[22]Kretzschmar, M.: A renewal equation with birth-death process as a model for parasitic infections. J. Math. Biol. 27, 191–221 (1989) · doi:10.1007/BF00276103
[23]Liu, L. P., Cheng, K. S.: Global stability of a predator-prey system. J. Math. Biol. 26, 65–71 (1988) · Zbl 0638.92017 · doi:10.1007/BF00280173
[24]MacDonald, G.: The analysis of equilibrium in malaria. Tropical Diseases Bulletin 49, 813–828 (1952)
[25]MacDonald, G.: The measurement of malaria transmission. Proc R. Soc. Medic. 48, 295–301 (1955)
[26]Bruce-Chwatt, L. J., Glanville, V. J. (eds.) Dynamics of tropical disease. Selected papers by G. Macdonald. Oxford: Oxford University Press 1973
[27]Mech, L. D., McRoberts, R. E., Peterson, R. O., Page, R. E.: Relationship of deer and moose populations to previous winter’s snow. J. Anim. Ecol. 56, 615–627 (1987) · doi:10.2307/5072
[28]Peterson, R. O., Page, R. E.: Wolf density as a predictor of predation rate. Swedish Wildlife Research Suppl. 1, 771–773 (1987)
[29]Peterson, R. O., Page R. E.: The rise and fall of isle Royale wolves, 1975–1986. J. Mamm. 69(I), 89–99 (1988) · doi:10.2307/1381751
[30]Rosenzweig, M. L.: Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science 171, 385–387 (1971) · doi:10.1126/science.171.3969.385
[31]Waldstätter, R., Hadeler, K. P., Greiner G.: A Lotka-McKendrick model for a population structured by the level of parasitic infection. SIAM J. Math. Anal. 19, 1108–1118 (1988) · Zbl 0659.92015 · doi:10.1137/0519075
[32]Waltman, P.: Deterministic threshold models in the theory of epidemics. (Lect. Notes Biomath. vol. 1) Berlin Heidelberg New York: Springer 1974