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Global stability of a predator-prey system with stage structure for the predator. (English) Zbl 1062.34056

The authors consider the following predator-prey model with stage structure \[ \begin{aligned} x^{\prime }(t)&=x(t)\left( r-ax(t)-\frac{by_2(t)}{1+mx(t)} \right) ,\\ y_1^{\prime }(t)&=\frac{kbx(t)y_2(t)}{1+mx(t)}-\left( D+v_1\right) y_1(t),\\ y_2^{\prime }(t)&=Dy_1(t)-v_2y_2(t),\end{aligned} \] where \(x(t)\) is the density of prey at time \(t\), \(y_1(t),y_2(t)\) are the densities of the immature and mature predators at time \(t\). Some feasible sufficient conditions are obtained for the global asymptotic stability of a positive steady by using the theory of competitive systems, compound matrices and stability of periodic orbits. Some known results are improved.

MSC:

34D23 Global stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C60 Qualitative investigation and simulation of ordinary differential equation models
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