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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.

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
92D30 Epidemiology
92D40 Ecology
34K21 Stationary solutions of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K19 Invariant manifolds of functional-differential equations
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References:
[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) · JFM 53.0517.01
[2] Lotka, A.J., Elements of physical biology, (1925), Williams & Wilkins Co. Baltimore · JFM 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) · JFM 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
[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
[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
[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, 4, 713-731, (2009) · Zbl 1168.92328
[8] Chattopadhyay, J.; Arino, O., A predator – prey model with disease in the prey, Nonlinear anal., 36, 747-766, (1999) · Zbl 0922.34036
[9] Xiao, Y.N.; Chen, L.S., Analysis of a three species eco-epidemiological model, J. math. anal. appl., 258, 2, 733-754, (2001) · Zbl 0967.92017
[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, 3, 855-869, (1992) · Zbl 0760.92018
[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, 2, 95-105, (1992) · Zbl 0777.92014
[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
[13] Aiello, W.G.; Freedman, H.I., A time-delay model of single-species growth with stage structure, Math. biosci., 101, 2, 139-153, (1990) · Zbl 0719.92017
[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
[15] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0425.34048
[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
[17] Song, X.; Chen, L., Optimal harvesting and stability for a two species competitive system with stage structure, Math. biosci., 170, 2, 173-186, (2001) · Zbl 1028.34049
[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
[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) · Zbl 1296.92102
[20] Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and application of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
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