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A predator-prey system with a stage structure for the prey. (English) Zbl 1132.92340
Summary: This paper considers a periodic predator-prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.

MSC:
92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
37N25 Dynamical systems in biology
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[1] Hofbauer, J.; Sigmund, K., Evolutionary games and population dynamics, (1998), Cambridge University Press Cambridge · Zbl 0914.90287
[2] Murray, J.D., Mathematical biology, (1993), Springer Heidelberg · Zbl 0779.92001
[3] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[4] Takeuchi, Y., Global dynamical properties of lotka – volterra systems, (1996), World Scientific Singapore · Zbl 0844.34006
[5] Teng, Z., Uniform persistence of the periodic predator – prey lotka – volterra systems, Appl. anal., 72, 339-352, (1999) · Zbl 1031.34045
[6] 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
[7] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage-structure population growth with state-dependent time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018
[8] 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
[9] Wang, W.; Chen, L., A predator – prey system with stage-structure for predator, Comput. math. appl., 33, 83-91, (1997)
[10] Bernard, O.; Souissi, S., Qualitative behavior of stage-structure populations: application to structural validation, J. math. biol., 37, 291-308, (1998) · Zbl 0919.92035
[11] 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
[12] Wang, S., Research on the suitable living environment of the rana temporaria chensinensis larva, Chinese J. zool., 32, 1, 38-41, (1997)
[13] Zhao, X.-Q., The qualitative analysis of N-species lotka – volterra periodic competition systems, Math. comput. modelling, 15, 3-8, (1991) · Zbl 0756.34048
[14] Cui, J.; Song, X., Permanence of predator – prey system with stage structure, Discrete contin. dyn. syst. ser. B, 4, 547-554, (2004) · Zbl 1100.92062
[15] Zhao, X.-Q., Dynamical systems in population biology, (2003), Springer-Verlag New York
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