×

Permanence and global attractivity of a delayed periodic logistic equation. (English) Zbl 1101.34058

Summary: We consider the delayed periodic logistic equation
\[ \dot N(t)=N(t)[a(t)-b(t)N^p(t-\sigma(t))-c(t)N^q(t-\tau(t))], \]
which describes the evolution of a single species. Sufficient conditions which guarantee the permanence and the globally attractivity of the system are obtained.

MSC:

34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yan, J.; Feng, Q., Global attractivity and oscillation in a nonlinear delay equation, Nonlinear Anal., 43, 1, 101-108 (2001) · Zbl 0987.34065
[2] Gopalsamy, K.; Ladas, G., On the oscillation and asymptotic behavior of \(\dot{N}(t) = N(t) [a + bN(t - \tau) - cN^2(t - \tau)]\), Quart. Appl. Math., 48, 3, 433-440 (1990) · Zbl 0719.34118
[3] Lalli, B. S.; Zhang, B. G., On a periodic delay population model, Quart. Appl. Math., 52, 1, 35-42 (1994) · Zbl 0788.92022
[4] Li, Y.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 1, 260-280 (2001) · Zbl 1024.34062
[5] Chen, Y. M., Periodic solution of a delayed periodic logistic equation, Appl. Math. Lett., 16, 1047-1051 (2003) · Zbl 1118.34327
[6] Teng, Z. D., Permanence and stability in non-autonomous logistic systems with infinite delay, Dyn. Syst., 17, 3, 187-202 (2002) · Zbl 1035.34086
[7] Cui, J. A.; Takeuchi, Y., Permanence of a single-species dispersal system and predator survival, J. Comput. Appl. Math., 175, 375-394 (2005) · Zbl 1058.92042
[8] Chen, F. D., On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180, 1, 33-49 (2005) · Zbl 1061.92058
[9] Fan, M.; Ye, D.; Wong, P. J.Y.; Agarwal, R. P., Periodicity in a class of non-autonomous scalar equations with deviating arguments and applications to population models, Dyn. Syst.: Int. J., 19, 3, 279-301 (2004) · Zbl 1062.34073
[10] Chen, F. D.; Lin, F. X.; Chen, X. X., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158, 1, 45-68 (2004) · Zbl 1096.93017
[11] Huo, H. F.; Li, W. T., Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput., 148, 1, 35-46 (2004) · Zbl 1057.34093
[12] Chen, F. D.; Lin, S. J., Periodicity in a logistic type system with several delays, Comput. Math. Appl., 48, 1-2, 35-44 (2004) · Zbl 1061.34050
[13] Chen, F. D., Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288, 1, 132-142 (2003)
[14] Chen, F. D., Persistence and periodic orbits for two-species non-autonomous diffusion Lotka-Voltrra models, Appl. Math. J. Chin. Univ. Ser. B, 19, 4, 359-366 (2004) · Zbl 1074.34053
[15] Chen, F. D., Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162, 3, 1279-1302 (2005) · Zbl 1125.93031
[16] Kuang, Y., Delay differential equations with application in population dynamics, Mathematics in Science and Engineering, vol. 191 (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[17] Teng, Z. D.; Lu, Z. Y., The effect of dispersal on single-species nonautonomous dispersal models with delays, J. Math. Biol., 42, 439-454 (2001) · Zbl 0986.92024
[18] Wang, W. D.; Chen, L. S., Global stability of a population dispersal in a two-patch environment, Dyn. Syst. Appl., 6, 207-216 (1997) · Zbl 0892.92026
[19] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments, Nonlinear Anal. Real World Appl., 5, 1, 183-206 (2004) · Zbl 1066.92059
[20] Hale, J., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0352.34001
[21] Barbălat, I., Systems d’equations differential d’oscillations nonlinearies, Rev. Roumaine Math. Pure Appl., 4, 2, 267-270 (1959) · Zbl 0090.06601
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.