# zbMATH — the first resource for mathematics

Permanence and stability of a predator-prey system with stage structure for predator. (English) Zbl 1117.34070
The authors propose the following delayed predator-prey system with stage structure for predator
$\frac{dx(t)}{dt}=x(t)\left( r-a(t-\tau _1)-by(t)\right) ,$
$\frac{dy(t)}{dt} =\alpha e^{-\gamma \tau _2}y(t-\tau _2)x(t-\tau _2)-vy(t)-\beta y^2(t),$
$\frac{dy_j(t)}{dt}=\alpha x(t)y(t)-\alpha e^{-\gamma \tau _2}y(t-\tau _2)x(t-\tau _2)-\gamma y_j(t),$
where $$x(t),y(t),y_j(t)$$ are the densities of prey, mature predator and immature predator at time $$t,$$ and all the parameters are positive. Some results concerning with the boundedness of solutions, permanence and stability of the equilibrium are established.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
##### Keywords:
permanence; stability
Full Text:
##### References:
 [1] Aiello, W.G.; Freedman, H.I., A time-delay model of single-species growth with stage structure, Math. biosci., 101, 139-153, (1990) · Zbl 0719.92017 [2] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. appl. math., 52, 3, 855-869, (1992) · Zbl 0760.92018 [3] Brauer, F.; Ma, Z., Stability of stage-structured population models, J. math. anal. appl., 126, 301-315, (1987) · Zbl 0634.92014 [4] Freedman, H.I.; Wu, J.H., Persistence and global asymptotic stability of single-species dispersal models with stage structure, Quart. appl. math., 49, 351-371, (1991) · Zbl 0732.92021 [5] Goh, B.S., Global stability in two species interactions, J. math. biol., 3, 313-318, (1976) · Zbl 0362.92013 [6] Hastings, A., Global stability in two species systems, J. math. biol., 5, 399-403, (1978) · Zbl 0382.92008 [7] He, X.Z., Stability and delays in a predator – prey system, J. math. anal. appl., 198, 355-370, (1996) · Zbl 0873.34062 [8] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press London · Zbl 0777.34002 [9] Kuang, Y.; So, J.W.H., Analysis of a delayed two-stage population model with space-limited recruitment, SIAM J. appl. math., 55, 6, 1675-1696, (1995) · Zbl 0847.34076 [10] Magnusson, K.G., Destabilizing effect of cannibalism on a structured predator – prey system, Math. biosci., 155, 61-75, (1999) · Zbl 0943.92030 [11] Saito, Y.; Takeuchi, Y., A time-delay model for prey – predator growth with stage structure, Canad. appl. math. quart., 11, 3, 293-302, (2003) · Zbl 1087.34551 [12] Song, X.; Chen, L., Optimal harvesting and stability for a two species competitive system with stage structure, Math. biosci., 170, 2, 173-186, (2001) · Zbl 1028.34049 [13] Wang, W.; Chen, L., A predator – prey with stage-structure for predator, Computer math. appl., 33, 83-91, (1997) [14] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085 [15] Wang, W.; Mulone, G.; Salone, F., Permanence and stability of a stage-structured predator – prey model, J. math. anal. appl., 262, 499-528, (2001) · Zbl 0997.34069 [16] Wright, E.M., A nonlinear differential difference equation, J. reine angew. math., 194, 66-87, (1955) · Zbl 0064.34203 [17] Xu, R.; Chen, L., Persistence and global stability for a delayed nonautonomous predator – prey system without dominating instantaneous negative feedback, J. math. anal. appl., 262, 50-61, (2001) · Zbl 0997.34070
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.