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On the asymptotic stability of an Hassell predator-prey model with mutual interference. (English) Zbl 1266.34085

The author studies the following Hassell predator-prey model with mutual interference
\[ \begin{aligned}\dot x(t)&=x(t)[r_1(t)-b_1(t)x(t)-c_1(t)y^m(t)],\\ \dot y(t)&=y(t)[-r_2(t)+c_2(t)x(t)y^{m-1}(t)-b_2(t)y(t)],\end{aligned}\tag{1} \] where \(x(t)\) and \(y(t)\) are the densities of the prey and the predator populations at time \(t\), the functions \(r_i(t), b_i(t), c_i(t)\;(i=1,2)\) are assumed to be continuous and defined on \([0, \infty)\), and are bounded above and below by positive constants.
The existence of an absorbing set is shown together with the global nonlinear asymptotic stability of the positive critical points for which the essential tool of the Lyapunov direct method for nonautonomous systems is used.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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