## Permanence of periodic Holling type-IV predator-prey system with stage structure for prey.(English)Zbl 1156.34327

Summary: We study the permanence of the following periodic Holling type-IV predator-prey system with stage structure for prey
\begin{aligned} \dot x_1(t)=a(t) & x_2(t)-b(t)x_1(t)-d(t)x^2_1(t)-\frac{p(t)x_1(t)}{k(t)+x_1^2(t)}y(t),\\ \dot x_2(t)=c(t) & x_1(t)-f(t)x^2_2(t),\\ \dot y(t)=y(t)& \left(-g(t)+\frac{h(t)x_1(t)}{k(t)+x_1^2(t)}-q(t)y(t)\right).\end{aligned}
Under certain assumptions, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general)

### Keywords:

nonautonomous; permanence; Holling type-IV; stage structure
Full Text:

### References:

 [1] Andrews, J.F., A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. bioeng., 10, 707-723, (1968) [2] Sokol, W.; Howell, J.A., Kinetics of phenol oxidation by washed cells, Biotechnol. bioeng., 23, 2039-2049, (1980) [3] Cui, J.A.; Song, X.Y., Permanence of a predator – prey system with stage structure, Discrete contin. dyn. syst. ser. B, 4, 3, 547-554, (2004) · Zbl 1100.92062 [4] Cui, J.A.; Takeuchib, Y., A predator – prey system with a stage structure for the prey, Math. comput. modelling, 44, 1126-1132, (2006) · Zbl 1132.92340 [5] Zhang, X.; Chen, L.; Neumann, A.U., The stage-structured predator – prey model and optimal harvesting policy, Math. biosci., 168, 201-210, (2000) · Zbl 0961.92037 [6] Cui, J.; Chen, L.; Wang, W., The effect of dispersal on population growth with stage-structure, Comput. math. appl., 39, 91-102, (2000) · Zbl 0968.92018 [7] Zhao, X.-Q., The qualitative analysis of N-species lotka – volterra periodic competition systems, Math. comput. modelling, 15, 3-8, (1991) · Zbl 0756.34048 [8] Zhao, X.-Q., Dynamical systems in population biology, (2003), Springer-Verlag New York
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.