zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

34C25Periodic solutions of ODE
34D23Global stability of ODE
34D40Ultimate boundedness (MSC2000)
37N25Dynamical systems in biology
Full Text: DOI
[1] Hofbauer, J.; Sigmund, K.: Evolutionary games and population dynamics. (1998) · Zbl 0914.90287
[2] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001
[3] Freedman, H. I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023
[4] Takeuchi, Y.: Global dynamical properties of Lotka--Volterra systems. (1996) · 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, No. 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) · Zbl 1023.37047