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.


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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