## 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
Full Text:

### References:

 [1] Bence, J. R., Nisbet, R. M.: Space limited recruitment in open systems: The importance of time delays. Ecology, 70, 1434–1441 (1989) [2] Aiello, W. G., Freedman, H. I.: A time delay model of single species growth with stage structure. Math. Biosci, 101, 139–156 (1990) · Zbl 0719.92017 [3] Wang, W., Chen, L.: A predator-prey system with stage-structure for predator. Computers Math. Applic., 33(8), 83–91 (1997) [4] Wang, W.: Global dynamics of a population model with stage structure for predator, in: L. Chen et al (Eds), Advanced topics in Biomathematics, Proceeding of the international conference on mathematical biology, World Scientific Publishing Co. Pte. Ltd., 253–257 (1997) · Zbl 0986.92026 [5] Magnusson, K. G.: Destabilizing effect of cannibalism on a structured predator-prey system. Math. Biosci, 155, 61–75 (1999) · Zbl 0943.92030 [6] Smith, H. L.: Systems of ordinary differential equations which generate an order preserving flow. SIAM Rev., 30, 87–98 (1988) · Zbl 0674.34012 [7] Hirsh, M. W.: Systems of differential equations which are competitive or cooperative, IV: structural stabilities in three dimensional systems, SIAM J. Math. Anal., 21, 1225–1234 (1990) · Zbl 0734.34042 [8] Verhulst, F.: Nonlinear differential equations and dynamical systems, Springer, Berlin (1990) · Zbl 0685.34002 [9] Hale, J. K.: Ordinary differential equations, Wiley, New York (1969) · Zbl 0186.40901 [10] Li, Y., Muldowney, J. S.: Global stability for the SEIR model in epidemiology. Math. Biosci, 125, 155–164 (1995) · Zbl 0821.92022 [11] Muldowney, J. S.: Compound matrices and ordinary differential equations. Rocky Mountain J. Math., 20, 857–872 (1990) · Zbl 0725.34049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.