# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. (English) Zbl 1147.92031

Summary: By using the Krasnoselskii cone fixed point theorem, we obtain a sufficient condition as well as a necessary condition for the existence of positive periodic solutions of the following generalized A. J. Nicholson’s [An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–25 (1954)] blowflies model:

${x}^{\text{'}}\left(t\right)=-\delta \left(t\right)x\left(t\right)+\sum _{i=1}^{m}{p}_{i}\left(t\right)x\left(t-{\tau }_{i}\left(t\right)\right){e}^{-{q}_{i}\left(t\right)x\left(t-{\tau }_{i}\left(t\right)\right)},\phantom{\rule{1.em}{0ex}}t\ge 0·$

In the degenerate case, i.e., where the coefficients and delays of the above equation are all constants, a sufficient and necessary condition for the existence of positive periodic solutions is obtained. Our results are completely new, and generalize and improve some results from the literature.

##### MSC:
 92D25 Population dynamics (general) 34K13 Periodic solutions of functional differential equations 34K60 Qualitative investigation and simulation of models
##### Keywords:
cone fixed point theorem
##### References:
 [1] Gurney, W. S.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies (revisited), Nature 287, 17-21 (1980) [2] Li, Wantong; Fan, Yonghong: Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson’s blowflies model, J. comput. Appl. math. 201, 55-68 (2007) · Zbl 1117.34065 · doi:10.1016/j.cam.2006.02.001 [3] Kocic, V. L.; Ladas, G.: Oscillation and global attractivity in the discrete model of Nicholson’s blowflies, Appl. anal. 38, 21-31 (1990) · Zbl 0715.39003 · doi:10.1080/00036819008839952 [4] Kulenovic, M. R. S.; Ladas, G.; Sficas, Y. S.: Global attractivity in Nicholson’s blowflies, Comput. math. Appl. 18, 925-928 (1989) [5] Kulenovic, M. R. S.; Ladas, G.; Sficas, Y. S.: Global attractivity in population dynamics, Appl. anal. 43, 109-124 (1992) [6] Nicholson, A. J.: An outline of the dynamics of animal populations, Austral. J. Zool. 2, 9-25 (1954) [7] Luo, Jiaowan: Global attractivity of generalized Nicholson’s blowflies model, J. hunan univ. 9, No. 1, 13-17 (1994) · Zbl 0864.34048 [8] Liu, Xilan; Li, Wantong: Existence and uniqueness of positive periodic solutions of functional differential equations, J. math. Anal. appl. 293, 28-39 (2004) · Zbl 1057.34094 · doi:10.1016/j.jmaa.2003.12.012 [9] Saker, S. H.; Agarwal, S.: Oscillation and global attractivity in a periodic Nicholson’s blowflies model, Math. comput. Modelling 35, 719-731 (2002) · Zbl 1012.34067 · doi:10.1016/S0895-7177(02)00043-2 [10] Xu, Wangwen; Li, Jingwen: Global attractivity of the model for the survival of red blood cells with several delays, Ann. differential equations 14, No. 2, 357-363 (1998) [11] Liu, Guirong; Zhao, Aimin; Yan, Jurang: Existence and global attractivity of unique positive periodic solution for a lasota–wazewska model, Nonlinear anal. 64, 1737-1746 (2006) · Zbl 1099.34064 · doi:10.1016/j.na.2005.07.022 [12] Guo, Dajun: Nonlinear functional analysis, (2001)