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 R. R. Vance and 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.


34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics (general)
35K20 Initial-boundary value problems for second-order parabolic equations
34K13 Periodic solutions to functional-differential equations


Zbl 0716.92016
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), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · 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., LIII, 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) · Zbl 1042.34571
[12] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[13] Lalli, B. S.; Zhang, B. G., On a periodic population model, Q. Appl. Math., LII, 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, 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, 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.