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Permanence and stability of an age-structured prey-predator system with delays. (English) Zbl 1187.34110
Summary: We consider the following plausible age-structured prey-predator interaction model:
\begin{aligned} & \frac{dx_j(t)}{dt}=\alpha x(t)-\gamma x_k(t)-\alpha e^{\gamma \tau}x(t-\tau),\\ & \frac{dx(t)}{dt}=\alpha e^{-\gamma\tau}x(t-\tau)-\mu_1x(t)-mx^2(t)-\beta x(t)y(t),\\ & \frac{dy(t)}{dt}=b\beta x(t-\sigma)y(t-\sigma)-\mu_2y(t)-\omega y^2(t),\end{aligned} \tag{1}
where $$x_j(t)$$ and $$x(t)$$ represent, respectively, the juvenile and adult prey densities at time $$t$$; $$y(t)$$ represents the predator density at time $$t, \alpha,\mu_1,\gamma,\mu_2,\beta,\tau,\sigma,m$$ and $$\omega$$ are positive constants.
Mathematical analysis of the model equations with regard to boundedness of solutions, permanence, and stability is performed. By using the persistence theory for infinite-dimensional systems, sufficient conditions for the permanence of the system are obtained. By constructing suitable Lyapunov functions and using an iterative technique, sufficient conditions are also obtained for the global asymptotic stability of the positive equilibrium of the model.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
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