# zbMATH — the first resource for mathematics

Periodicity in a logistic type system with several delays. (English) Zbl 1061.34050
Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following logistic-type system with several delays $\frac{du(t)}{dt}=u(t)\left[a(t)-\sum^m_{j=1}b_j(t) \biggl(u\bigl(t-\tau_j(t)\bigr)\biggr)^{\theta_j} \right],$ where $$a(t)$$, $$b_j(t)$$ are positive periodic continuous functions with periodic $$\omega>0$$, $$\tau_j(t)$$ are nonnegative continuous periodic functions with periodic $$\omega>0$$. After that, by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the stability of the positive periodic solutions are obtained.

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations
Full Text:
##### References:
 [1] Gopalsamy, K., () [2] Gyori, I.; Ladas, G., Oscillation theory of delay differential equations, (1991), Oxford Science Publications Dordrecht · Zbl 0780.34048 [3] Lenhart, S.M.; Travis, C.C., Global stability of a biological model with time delay, (), 75-78 · Zbl 0602.34044 [4] Kuang, Y., () [5] Yu, J.S., Global attractivity of zero solution for a class of functional equations and its applications, Science in China, series A, 26, 23-33, (1996), (in Chinese) [6] Cao, Y.L.; Thomas, C.G., Ultimate bounds and global asymptotic stability for differential delay equations, Rocky mountain journal of mathematics, 25, 119-131, (1995) · Zbl 0829.34066 [7] Li, Y.K., Global attractivity in a periodic delay single species model, System science and complexity, 13, 1-6, (2000) · Zbl 0969.92022 [8] Li, Y.K., On a periodic logistic equation with several delay, Advances in mathematics, 28, 135-142, (1999) · Zbl 1054.34513 [9] Chen, Y.M., Periodic solution of a delayed periodic logistic equation, Appl. math. lett., 16, 7, 1047-1051, (2003) · Zbl 1118.34327 [10] 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 [11] Yan, J.R.; Feng, Q.X., Global existence and oscillation in a nonlinear delay equation, Nonlinear analysis, 43, 101-108, (2001) · Zbl 0987.34065 [12] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Boston, MA · Zbl 0326.34021 [13] Gui, Z.; Chen, L., Persistence and periodic solutions of a periodic logistic equation with time delays, Journal of mathematical research and exposition, 23, 1, 109-114, (2003), (in Chinese) · Zbl 1038.34080
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.