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)
Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure. (English) Zbl 1215.34102

This paper deals with an epidemiological predator-prey model in which the population of preys is divided into two classes: susceptible and infected prey, and the population of predators is divided into two stage groups: juveniles and adults. The age of maturity introduces a constant delay which leads to a nonlinear system of delayed differential equations.

Sufficient conditions for the asymptotic stability of the five equilibria are established. Considering the delay as a varying parameter, the stability of the positive equilibrium is analyzed. Under some conditions, a Hopf bifurcation occurs when the delay passes through a critical value. Using the normal form and center manifold theory, explicit formulae determining the direction of the bifurcating periodic solutions are provided.

The paper ends with some numerical simulations to illustrate the theoretical results.

34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
34K21Stationary solutions of functional-differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
[1]Kermack, W. O.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemics (part I), Proc. roy. Soc. ser. A 115, 700-721 (1927) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[2]Lotka, A. J.: Elements of physical biology, (1925) · Zbl 51.0416.06
[3]Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. R. Accad. naz. Dei lincei. Ser. VI 2, 31-113 (1926) · Zbl 52.0450.06
[4]Cheng, K.; Hus, S.; Lin, S.: Some results on a global stability of a predator–prey system, J. math. Biol. 12, 115-126 (1981) · Zbl 0464.92021 · doi:10.1007/BF00275207
[5]Kuang, Y.; Freedman, H. I.: Uniqueness of limit cycles in Gauss-type models of predator prey systems, Math. biosci. 88, 67-84 (1988) · Zbl 0642.92016 · doi:10.1016/0025-5564(88)90049-1
[6]Ruan, S. G.; Xiao, D. M.: Global stability in predator prey system with non-monotonic functional response, SIAM J. Appl. math. 61, 1445-1472 (2000) · Zbl 0986.34045 · doi:10.1137/S0036139999361896
[7]Zhou, X. Y.; Shi, X. Y.; Song, X. Y.: Analysis of a delay prey–predator model with disease in the prey species only, J. korean math. Soc. 46, No. 4, 713-731 (2009) · Zbl 1168.92328 · doi:10.4134/JKMS.2009.46.4.713
[8]Chattopadhyay, J.; Arino, O.: A predator–prey model with disease in the prey, Nonlinear anal. 36, 747-766 (1999) · Zbl 0922.34036 · doi:10.1016/S0362-546X(98)00126-6
[9]Xiao, Y. N.; Chen, L. S.: Analysis of a three species eco-epidemiological model, J. math. Anal. appl. 258, No. 2, 733-754 (2001) · Zbl 0967.92017 · doi:10.1006/jmaa.2001.7514
[10]Aiello, W. G.; Freedman, H. I.; Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. math. 52, No. 3, 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[11]Cao, Y.; Fan, J.; Gard, T. C.: The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear anal. 19, No. 2, 95-105 (1992) · Zbl 0777.92014 · doi:10.1016/0362-546X(92)90113-S
[12]Song, X.; Chen, L.: Modelling and analysis of a single species system with stage structure and harvesting, Math. comput. Modelling 36, 67-82 (2002) · Zbl 1024.92015 · doi:10.1016/S0895-7177(02)00104-8
[13]Aiello, W. G.; Freedman, H. I.: A time-delay model of single-species growth with stage structure, Math. biosci. 101, No. 2, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[14]Gourley, Stephen A.; Kuang, Yang: A stage structured predator–prey model and its dependence on maturation delay and death rate, J. math. Biol. 49, 188-200 (2004) · Zbl 1055.92043 · doi:10.1007/s00285-004-0278-2
[15]Hale, J. K.: Theory of functional differential equations, (1977)
[16]Yang, X.; Chen, L. S.; Chen, J. F.: Permanence and positive periodic solution for single-species nonautonomous delay diffusive model, Comput. math. Appl. 32, 109-116 (1996) · Zbl 0873.34061 · doi:10.1016/0898-1221(96)00129-0
[17]Song, X.; Chen, L.: Optimal harvesting and stability for a two species competitive system with stage structure, Math. biosci. 170, No. 2, 173-186 (2001) · Zbl 1028.34049 · doi:10.1016/S0025-5564(00)00068-7
[18]Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator–prey system, J. math. Biol. 36, 389-406 (1998) · Zbl 0895.92032 · doi:10.1007/s002850050105
[19]Adimy, M.; Crauste, F.; Ruan, S.: Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, B. math. Biol. 68, 2321-2351 (2006)
[20]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and application of Hopf bifurcation, (1981)