# 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)
Periodic solutions of delayed Leslie-Gower predator-prey models. (English) Zbl 1060.34039
Authors’ abstract: With the help of continuation theorem based on Gaines and Mawhin’s conincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for both the first and the second Leslie-Gower predator-prey models with distributed or state dependent delay. In the reviewer’s opinion, a relevant reference is [{\it Q. Liu} and {\it H. Zhou}, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 5, Paper No. 89 (2003; Zbl 1069.34063)] where coincidence degree is used to prove existence and global attractivity of periodic solutions in an $n$-species food chain system with time delays.

##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Fan, M.; Wang, K.: Periodicity in a delayed ratio-dependent predator--prey system. J. math. Anal. appl. 262, 179-190 (2001) · Zbl 0994.34058 [2] Freedman, H. I.; Wu, J.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. anal. 23, 689-701 (1992) · Zbl 0764.92016 [3] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031 [4] Gopalsamy, K.; Kulenovic, M. R. S.; Ladas, G.: Environmental periodicity and time delays in a ”food-limited” population model. J. math. Anal. appl. 147, 545-555 (1990) · Zbl 0701.92021 [5] Huo, H. F.; Li, W. T.: Periodic solution of a periodic two-species competition model with delays. Int. J. Appl. math. 12, No. 1, 13-21 (2003) · Zbl 1043.34074 [6] Korobeinikov, A.: A Lyapunov function for Leslie--gower predator--prey models. Appl. math. Lett. 14, 697-699 (2001) · Zbl 0999.92036 [7] Kuang, Y.: Delay differential equations with application in population dynamics. (1993) · Zbl 0777.34002 [8] Li, Y. K.: Periodic solutions of a periodic delay predator--prey system. Proc. am. Math. soc. 127, 1331-1335 (1999) · Zbl 0917.34057 [9] Li, Y. K.; Kuang, Y.: Periodic solutions of periodic delay Lotka--Volterra equations and systems. J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062 [10] Leslie, P. H.: Some further notes on the use of matrices in population mathematics. Biometrika 35, 213-245 (1948) · Zbl 0034.23303 [11] Leslie, P. H.: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45, 16-31 (1958) · Zbl 0089.15803 [12] May, R. M.: Stability and complexity in model ecosystems. (1974) [13] Pielou, E. C.: Mathematical ecology. (1977) · Zbl 0259.92001 [14] Smith, H. L.; Kuang, Y.: Periodic solutions of delay differential equations of threshold-type delay. Oscillation and dynamics in delay equations, contemporary mathematics 129, 153-176 (1992) · Zbl 0762.34044 [15] Tang, B. R.; Kuang, Y.: Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems. Tohoku math. J. 49, 217-239 (1997) · Zbl 0883.34074 [16] Zhang, B. G.; Gopalsamy, K.: Global attractivity and oscillations in a periodic delay-logistic equations. J. math. Anal. appl. 150, 274-283 (1990) · Zbl 0711.34090 [17] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator--prey systems. Nonlinear anal. TMA 28, 1373-1394 (1997) · Zbl 0872.34047