## Permanence of periodic Holling type predator–prey system with stage structure for prey.(English)Zbl 1111.34039

Summary: We study the permanence of the following periodic Holling-type predator-prey system with stage structure for prey \begin{aligned} \dot x_1(t) & =a(t)x_2(t)-b(t)x_1(t) -d(t)x^2_1(t)-\frac{e(t)x^\gamma_1(t)} {p(t)+ x^\gamma_1(t)}y(t),\\ \dot x_2(t)& =c(t)x_1(t)-f(t)x^2_2(t),\\ \dot y(t) &=y(t)\left(-g(t)+\frac{h(t)x_1^\gamma(t)} {p(t)+x^\gamma_1(t)}-q(t)y(t)\right).\end{aligned} A sufficient and necessary condition which guarantees the predator and the prey species to be permanent is obtained. Some new results are obtained.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D25 Population dynamics (general)
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### References:

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