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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
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