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Permanence and stability of a predator-prey system with stage structure for predator. (English) Zbl 1117.34070
The authors propose the following delayed predator-prey system with stage structure for predator
$\frac{dx(t)}{dt}=x(t)\left( r-a(t-\tau _1)-by(t)\right) ,$
$\frac{dy(t)}{dt} =\alpha e^{-\gamma \tau _2}y(t-\tau _2)x(t-\tau _2)-vy(t)-\beta y^2(t),$
$\frac{dy_j(t)}{dt}=\alpha x(t)y(t)-\alpha e^{-\gamma \tau _2}y(t-\tau _2)x(t-\tau _2)-\gamma y_j(t),$
where $$x(t),y(t),y_j(t)$$ are the densities of prey, mature predator and immature predator at time $$t,$$ and all the parameters are positive. Some results concerning with the boundedness of solutions, permanence and stability of the equilibrium are established.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
##### Keywords:
permanence; stability
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