zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Analysis of a three species eco-epidemiological model. (English) Zbl 0967.92017
Summary: This paper formulates and analyzes a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, permanence, and global stability are analyzed. It is also shown that for some parameter values, the positive equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation. A concluding discussion with numerical simulation is then presented.

37N25Dynamical systems in biology
34D05Asymptotic stability of ODE
34D23Global stability of ODE
34C23Bifurcation (ODE)
Full Text: DOI
[1] Hethcote, H. W.: A thousand and one epidemic models. Lecture notes in biomathematics 100, 504-515 (1994) · Zbl 0819.92020
[2] Anderso, R. M.; May, R. M.: Infectious disease of humans, dynamics and control. (1991)
[3] Bailey, N. J. T.: The mathematical theory of infectious disease and its applications. (1975) · Zbl 0334.92024
[4] Diekmann, O.; Hecsterbeck, J. A. P.; Metz, J. A. J.: The legacy of kermack and mckendrick. Epidemic models: their structure and relation to data (1994)
[5] Hadeler, K. P.; Freedman, H. I.: Predator--prey population with parasitic infection. J. math. Biol. 27, 609-631 (1989) · Zbl 0716.92021
[6] Chattopadhyay, J.; Arino, O.: A predator--prey model with disease in the prey. Nonlinear anal. 36, 749-766 (1999) · Zbl 0922.34036
[7] Venturino, E.: The influence of disease on Lotka-Volterra systems. Rockymount. J. Math. 24, 389-402 (1994) · Zbl 0799.92017
[8] J. C. Holmes and W. M. Bethel, Modification of intermediate host behavior by parasites, inBehavioural Aspects of Parasite Transmission (E. V. Canning and C. A. Wright, Eds.), Zool. f.Linnean Soc.51 Suppl. 1 1972, 123--149.
[9] Peterson, R. O.; Page, R. E.: Wolf density as a predictor of predation rate. Swedish wildlife research suppl. 1, 771-773 (1987)
[10] Chen, L. S.; Chen, J.: Nonlinear biological dynamic systems. (1993)
[11] Anderson, R. M.; May, R. M.: The population dynamics of microparasites and their intervebrates hosts. Proc. roy. Soc. lond. B 291, 451-463 (1981)
[12] Nagumo, N.: Uber die lage der integralkurven gewonlicher differantialgleichungen. Proc. phys. Math. soc. Japan 24, 551-567 (1942) · Zbl 0061.17204
[13] Liu, W. M.: Criterion of Hopf bifurcation without using eigenvalues. J. math. Anal. appl. 182, 250-255 (1994) · Zbl 0794.34033
[14] Freedman, H. I.; Rao, V. Sree Hari: The trade-off between mutual interference and time lags in predator--prey-systems. Bull. math. Biol. 45, 991-1003 (1983) · Zbl 0535.92024
[15] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[16] Kuang, Y.: Delay differential equations. (1993) · Zbl 0777.34002
[17] Hale, J. K.; Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. anal. 20, 388-396 (1989) · Zbl 0692.34053