# 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 and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments. (English) Zbl 1186.34118

Summary: The model discussed in this paper is described by the following periodic 3-species Lotka-Volterra predator-prey system with several deviating arguments:

$\left\{\begin{array}{c}{x}_{1}^{\text{'}}\left(t\right)={x}_{1}\left(t\right)\left({r}_{1}\left(t\right)-{a}_{11}\left(t\right){x}_{1}\left(t-{\tau }_{11}\left(t\right)\right)-{a}_{12}\left(t\right){x}_{2}\left(t-{\tau }_{12}\left(t\right)\right)-{a}_{13}\left(t\right){x}_{3}\left(t-{\tau }_{13}\left(t\right)\right)\right)\hfill \\ {x}_{2}^{\text{'}}\left(t\right)={x}_{2}\left(t\right)\left({r}_{2}\left(t\right)-{a}_{21}\left(t\right){x}_{1}\left(t-{\tau }_{21}\left(t\right)\right)-{a}_{22}\left(t\right){x}_{2}\left(t-{\tau }_{22}\left(t\right)\right)-{a}_{23}\left(t\right){x}_{3}\left(t-{\tau }_{23}\left(t\right)\right)\right)\hfill \\ {x}_{3}^{\text{'}}\left(t\right)={x}_{3}\left(t\right)\left({r}_{2}\left(t\right)-{a}_{31}\left(t\right){x}_{1}\left(t-{\tau }_{31}\left(t\right)\right)-{a}_{32}\left(t\right){x}_{2}\left(t-{\tau }_{32}\left(t\right)\right)-{a}_{33}\left(t\right){x}_{3}\left(t-{\tau }_{33}\left(t\right)\right)\right)\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(*\right)$

where ${x}_{1}\left(t\right)$ denotes the density of prey species at time $t$, ${x}_{2}\left(t\right)$ and ${x}_{3}\left(t\right)$ denote the density of predator species at time $t$, ${r}_{i},{a}_{ij}\in C\left(ℝ,\left[0,\infty \right)\right)$ and ${\tau }_{ij}\in C\left(ℝ,ℝ\right)$ are $w$-periodic functions with

${\overline{r}}_{i}=\frac{1}{w}{\int }_{0}^{w}{r}_{i}\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds>0;\phantom{\rule{1.em}{0ex}}{\overline{a}}_{ij}=\frac{1}{w}{\int }_{0}^{w}{a}_{ij}\left(s\right)>0,\phantom{\rule{1.em}{0ex}}I,j=1,2,3·$

By using Krasnoselskii’s fixed point theorem and the construction of Lyapunov function, a set of easily verifiable sufficient conditions are derived for the existence and global attractivity of positive periodic solutions of (*).

##### MSC:
 34K60 Qualitative investigation and simulation of models 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general)
##### References:
 [1] Alvarez, C.; Lazer, A. C.: An application of topological degree to the periodic competing species model, J. aust. Math. soc. Ser. B 28, 202-219 (1986) · Zbl 0625.92018 · doi:10.1017/S0334270000005300 [2] Ahmad, S.: On the nonautoomous Lotka–Volterra competition equations, Proc. amer. Math. soc. 117, 199-204 (1993) · Zbl 0848.34033 · doi:10.2307/2159717 [3] Battaaz, A.; Zanolin, F.: Coexistence states for periodic competition Kolmogorov systems, J. math. Anal. appl. 219, 179-199 (1998) · Zbl 0911.34037 · doi:10.1006/jmaa.1997.5726 [4] Chen, Y.; Zhou, Z.: Stable periodic solution of a discrete periodic Lotka–Volterra competition system, J. math. Anal. appl. 277, 358-366 (2003) · Zbl 1019.39004 · doi:10.1016/S0022-247X(02)00611-X [5] Cushing, J. M.: Two species competition in a periodic environment, J. math. Biol. 10, 385-400 (1980) · Zbl 0455.92012 · doi:10.1007/BF00276097 [6] Fan, M.; Wang, K.: Global periodic solutions of a generalized n-species gilpn–ayala competition model, Comput. math. Appl. 40, 1141-1151 (2000) · Zbl 0954.92027 · doi:10.1016/S0898-1221(00)00228-5 [7] Fan, M.; Wang, K.; Jiang, D. Q.: Existence and global attractivity of positive peridic solutions of n-species Lotka–Volterra competition systems with several deviating arguments, Math. biosci. 160, 47-61 (1999) · Zbl 0964.34059 · doi:10.1016/S0025-5564(99)00022-X [8] Gopalsamy, K.: Global asymptotical stability in a periodic Lotka–Volterra system, J. aust. Math. soc. Ser. B 29, 66-72 (1985) · Zbl 0588.92019 · doi:10.1017/S0334270000004768 [9] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002 [10] Li, Y. K.: Periodic solutions of N-species competition system with delays, J. biomath. 12, 1-12 (1997) · Zbl 0891.92027 [11] Li, Y. K.: On a periodic delay logistic type population model, Ann. differential equations 14, 29-36 (1998) · Zbl 0966.34067 [12] Li, Y. K.: Periodic solutions for delay Lotka–Volterra competition systems, J. math. Anal. appl. 255, 260-280 (2001) [13] 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 · doi:10.1006/jmaa.2000.7248 [14] Tang, H. X.; Zou, F. X.: On positive periodic solutions of Lotka–Volterra competition systems with deviating arguments, Proc. amer. Math. soc.. 134, 2967-2974 (2006) · Zbl 1101.34056 · doi:10.1090/S0002-9939-06-08320-1 [15] Tang, X. H.; Zou, X. F.: Global attractivity of positive periodic solution to periodic Lotka–Volterra competition systems with pure delay, J. differential equations 228, 580-610 (2006) · Zbl 1113.34052 · doi:10.1016/j.jde.2006.06.007 [16] Krasnoselskii, M. A.: Positive solutions of operator equations, noordhoff, Groningen, (1964) · Zbl 0121.10604 [17] Roydin, H. L.: Real analysis, (1998)