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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
##### Keywords:
eco-epidemiology; delayed stage structure; Hopf bifurcation
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##### References:
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