# 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 neutral logarithmic population model with multiple delays. (English) Zbl 1061.34053
The authors study the existence of positive periodic solutions for a neutral delay logarithmic population model with multiple delays of the form $${dN\over dt}= N(t)\Biggl[r(t)- \sum^N_{j=1} a_i(t)\ln N(t- \sigma_i(t))- \sum^m_{j=1} b_j(t){d\over dt}\ln N(t-\tau_j(t))\Biggr],\tag1$$ with $\sigma_j(t)\ge 0$ and $\tau_j(t)\ge 0$. The goal of this paper is to establish some criteria guaranteeing the existence of positive periodic solutions of (1). Under some suitable assumptions on the data of (1), the authors obtain a new existence result by using an abstract continuation theorem for $k$-set contraction and some other analytic techniques.

##### MSC:
 34K13 Periodic solutions of functional differential equations 35K40 Systems of second-order parabolic equations, general 34K60 Qualitative investigation and simulation of models 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [2] R.E. Gaines, J.L. Mawhin, Lectures Notes in Mathematics, Vol. 568, Springer, Berlin, 1977. [3] Gopalsamy, K.; He, X.; Wen, L.: On a periodic neutral logistic equation. Glasgow math. J. 33, 281-286 (1991) · Zbl 0737.34050 [4] Gopalsamy, K.; Zhang, B. G.: On a neutral delay logistic equation. Dyn. stability systems 2, 183-195 (1988) · Zbl 0665.34066 [5] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002 [6] Kuang, Y.; Feldstein, A.: Boundedness of solutions of a nonlinear nonautonomous neutral delay equation. J. math. Anal. appl. 156, 293-304 (1991) · Zbl 0731.34089 [7] Liu, Z. D.; Mao, Y. P.: Existence theorem for periodic solutions of higher order nonlinear differential equations. J. math. Anal. appl. 216, 481-490 (1997) · Zbl 0892.34040 [8] Petryshynand, W. V.; Yu, Z. S.: Existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems. Nonlinear anal. 6, No. 9, 943-969 (1982) · Zbl 0525.34015 [9] Pielou, E. C.: Mathematics ecology. (1977) · Zbl 0259.92001 [10] Yang, Z.; Cao, J.: Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models. Appl. math. Comput. 142, 123-142 (2003) · Zbl 1037.34066