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)
An impulsive predator-prey model with communicable disease in the prey species only. (English) Zbl 1162.92043
Summary: A system of impulsive differential equations describing predator-prey dynamics with impulsive effect is proposed and analyzed with the assumption that a transmissible disease is spreading among the prey species only. At first, the “semi-trivial” periodic solution $(S(t),0,0)$ is given. After that, the existence of an “infection-free” periodic solution $(S(t),0,P(t))$ and the “predator-free” periodic solution have been obtained via bifurcations. Finally, the method of coincidence degree has been used to derive a set of sufficient conditions for the existence of at least one strictly positive periodic solution. Numerical simulations and a brief discussion conclude the paper.

MSC:
 92D40 Ecology 34A37 Differential equations with impulses 92D30 Epidemiology 34C25 Periodic solutions of ODE
Full Text:
References:
 [1] Hadeler, K. P.; Freedman, H. I.: Predator-prey population with parasite infection. J. math. Biol. 27, 609-631 (1989) · Zbl 0716.92021 [2] Freedman, H. I.: A model of predator--prey dynamics as modified by the action of a parasite. Math. biosci. 99, 143-155 (1990) · Zbl 0698.92024 [3] Beltrami, E.; Carroll, T. O.: Modelling the role of viral disease in recurrent phytoplankton blooms. J. math. Biol. 32, 857-863 (1994) · Zbl 0825.92122 [4] Haque, M.; Venturino, E.: An ecoepidemiological model with disease in predator: the ratio-dependent case. Math. methods appl. Sci. 30, No. 14, 1791-1809 (2006) · Zbl 1126.92050 [5] Venturino, E.: Epidemics in predator--prey model: disease in the predators. IMA J. Math. appl. Med. biol. 19, 185-205 (2002) · Zbl 1014.92036 [6] Xiao, Y.; Chen, L.: Modelling and analysis of a predator--prey model with disease in the prey. Math. biosci. 171, 59-82 (2001) · Zbl 0978.92031 [7] Xiao, Y.; Chen, L.: Analysis of a three species eco-epidemiological model. J. math. Anal. appl. 258, No. 2, 733-754 (2001) · Zbl 0967.92017 [8] Hethcote, H. W.; Wang, W.; Han, L.; Ma, Z.: A predator--prey model with infected prey. Theo. pop. Biol. 66, 259-268 (2004) [9] Liu, X. Z.: Practical stabilization of control systems with impulsive effects. J. math. Anal. appl. 166, 563-576 (1992) · Zbl 0757.93073 [10] Liu, X. Z.; Willms, A.: Impulsive stabilizability of autonomous systems. J. math. Anal. appl. 187, 17-39 (1994) · Zbl 0814.34010 [11] Tang, Sanyi; Chen, Lansun: The periodic predator--prey Lotka--Volterra model with impulsive effect. J. mech. Medicine biol. 2, 267-296 (2002) [12] Liu, Bing; Zhang, Yujuan; Chen, Lansun: The dynamical behaviors of a Lotka--Volterra predator--prey model concerning integrated pest management. Nonlinear anal. RWA 6, 227-243 (2005) · Zbl 1082.34039 [13] Stone, L.; Shulgin, B.; Agur, Z.: Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. comput. Modelling 31, 207-215 (2000) · Zbl 1043.92527 [14] D’onofrio, Alberto: Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures. Math. comput. Modelling 36, 473-489 (2002) · Zbl 1025.92011 [15] D’onofrio, Alberto: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. biosci. 179, 57-72 (2002) · Zbl 0991.92025 [16] Berthier, K.; Langlasis, M.; Auger, P.; Pontier, D.: Dynamics of a feline virus with two transmission models with exponentially growing host populations. Proc. roy. Soc. lond. B 267, 2049-2056 (2000) [17] De Jong, M. C. M; Diekmann, O.; Heesterbeek, J. A. P.: How does infection depend on the population size?. Epidemic models, their structure and relation in data (1994) [18] Hamilton, W. D.; Axelrod, R.; Tanese, R.: Sexual reproduction as an adaptation to resist parasite (a review). Proc. natl. Acad. sci. USA 87, 3566-3573 (1990) [19] Hammond, A. M.; Hardy, T. N.: Quality of diseased plants as host for insects. Plant strees--insect interactions, 381-432 (1989) [20] Holmes, J. C.; Bethel, W. M.: Modifications of intermediate host behaviour by parasite. Suppl I to zool. F. linnean soc. 51, 123-149 (1972) [21] Lakmeeh, A.; Arino, O.: Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam. contin. Discrete impulsive systems 7, 265-287 (2000) [22] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031 [23] Greenhalgh, D.; Haque, M.: A predator--prey model with disease in prey species only. Math. methods appl. Sci. 30, 911-929 (2007) · Zbl 1115.92049 [24] Haque, M.; Venturino, E.: Increase of the prey May decrease the healthy predator population in presence of a disease in the predator. Hermis j. 7, 38-59 (2006) · Zbl 1260.92116 [25] Jin, Z.; Haque, M.: Global stability analysis of an eco-epidemiological model of the salton sea. J. biol. Systems 14, No. 3, 373-385 (2006) · Zbl 1116.92065 [26] Venturino, E.: The influence of diseases on Lotka--Volterra systems. Rocky mountain J. Math. 24, 381-402 (1994) · Zbl 0799.92017 [27] Haque, M.; Venturino, E.: An eco-epidemiological model with disease in predator: the ratio-dependent case. Math. methods appl. Sci. 30, 1791-1809 (2007) · Zbl 1126.92050 [28] Gonzalez, M. R.; Hart, C. M.; Verfailile, J. R.; Hurlbert, S. H.: Salinity and fish effects on salton sea microecosystems: water chemistry and nutrient cycling. Hydrobiologia 381, 105-128 (1988)