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)
Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays. (English) Zbl 1119.34056
The author studies a nonautonomous single species Kolmogorov system with delays of the type $$\frac{dx(t)}{dt} = x(t)f(t,x_t), \ t \in [0,\infty),$$ where $x_t (s) = x(t+s)$ $\forall \ s \in [-\tau,0].$ This equation includes many special delayed nonautonomous population growth models of a single species. The results obtained in this paper extend the main results given by {\it R. R. Vance} and {\it E. A. Coddington} [J. Math. Biol. 27, 491--506 (1989; Zbl 0716.92016)], establishing in particular different general criteria on the boundedness, persistence, permanence, global asymptotic stability and the existence of positive periodic solutions for the equation above.

34K25Asymptotic theory of functional-differential equations
92D25Population dynamics (general)
35K20Second order parabolic equations, initial boundary value problems
34K13Periodic solutions of functional differential equations
Full Text: DOI
[1] Cao, Y.; Gard, T. C.: Ultimate bounds and global asymptotic stability for differential delay equations. Rocky mount. J. math. 25, 119-131 (1995) · Zbl 0829.34066
[2] Chen, Y.: Periodic solutions of a delayed periodic logistic equation. Appl. math. Lett. 16, 1047-1051 (2003) · Zbl 1118.34327
[3] Faria, T.: Global attractivity in scalar delayed differential equations with applications to population models. J. math. Anal. appl. 289, 35-54 (2004) · Zbl 1054.34122
[4] Freedman, H. I.; Wu, J. H.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. anal. 23, 689-701 (1992) · Zbl 0764.92016
[5] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[6] Gopalsamy, K.; He, X.: Dynamics of an almost periodic logistic integrodifferential equation. Meth. appl. Anal. 2, 38-66 (1995) · Zbl 0835.45004
[7] 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
[8] Gopalsamy, K.; Lalli, B. S.: Oscillatory and asymptotic behavior of a multiplicative delay logistic equation. Dyn. stab. Syst. 7, 35-42 (1992) · Zbl 0764.34049
[9] Grace, S. R.; Gyori, I.; Lalli, B. S.: Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation. Quart. appl. Math. 53, 69-79 (1995) · Zbl 0837.34073
[10] Grove, E. A.; Ladas, G.; Qian, C.: Global attractivity in a food-limited population model. Dyn. sys. Appl. 2, 243-249 (1993) · Zbl 0787.34061
[11] Gyori, I.: A new approach to the global stability problem in a delay Lotka -- Volterra differential equation. Math. comput. Modelling 31, 9-28 (2000)
[12] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[13] Lalli, B. S.; Zhang, B. G.: On a periodic population model. Q. appl. Math. 52, 35-42 (1994) · Zbl 0788.92022
[14] Li, Y.: The existence and global attractivity of periodic positive solutions for a class of delayed differential equations. Sci. China (Series A) 28, 108-118 (1998)
[15] Li, Y.; Kuang, Y.: Periodic solutions of periodic delay Lotka -- Volterra equations and systems. J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062
[16] Seifert, G.: Almost periodic solutions for delay logistic equation with almost periodic time dependence. Differential integral equations 9, 335-342 (1996) · Zbl 0838.34083
[17] So, J. W. -H.; Yu, J. S.: On the uniform stability for a food-limited population model with time delay. Proc. R. Soc. Edinburgh, sect. A 125, 991-1002 (1995) · Zbl 0844.34079
[18] Teng, Z.: Permanence and stability in non-autonomous logistic systems with infinite delay. Dynamical syst. 17, 187-202 (2002) · Zbl 1035.34086
[19] Teng, Z.; Chen, L.: The positive periodic solutions of periodic Kolmogorov type systems with delays. Acta math. Appl. sin. 22, 456-464 (1999) · Zbl 0976.34063
[20] Teng, Z.; Lu, Z.: The effect of dispersal on single-species nonautonomous dispersal models with delays. J. math. Biol. 42, 439-454 (2001) · Zbl 0986.92024
[21] Vance, R. R.; Coddington, E. A.: A nonautonomous model of population growth. J. math. Biol. 27, 491-506 (1989) · Zbl 0716.92016
[22] Yang, X.; Chen, L.; Chen, J.: Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models. Comput. math. Appl. 32, No. 4, 109-116 (1996) · Zbl 0873.34061
[23] Zhang, J.; Chen, L.: Periodic solutions of single-species nonautonomous diffusion models with continuous time delays. Math. comput. Modelling 23, No. 7, 17-27 (1996) · Zbl 0864.60058
[24] Zhang, B. G.; Gopalsamy, K.: Global attractivity and oscillations in a periodic delay-logistic equation. J. math. Anal. appl. 150, 274-283 (1990) · Zbl 0711.34090
[25] Zhao, C. J.; Debnath, L.; Wang, K.: Positive periodic solutions of a delayed model in population. Appl. math. Lett. 16, 561-565 (2003) · Zbl 1058.34088