## Existence and global stability of positive periodic solution of periodic integro-differential systems with feedback controls.(English)Zbl 0962.45003

The purpose of this paper is to derive a set of sufficient conditions for the existence and global asymptotic stability of a positive periodic solution of some periodic integro-differential systems with feedback controls, applied to describe the temporal evolution of $$n$$ competitive species population in a common environment, by using the technique of coincidence degree theory and Lyapunov functionals.

### MSC:

 45M05 Asymptotics of solutions to integral equations 92D25 Population dynamics (general) 45J05 Integro-ordinary differential equations 45M20 Positive solutions of integral equations 34K50 Stochastic functional-differential equations 93B52 Feedback control 45M10 Stability theory for integral equations 45M15 Periodic solutions of integral equations
Full Text:

### References:

 [1] Golpalsamy, K., Time lags and global stability in two species competition, Bull. math. biol., 42, 729-737, (1980) · Zbl 0453.92014 [2] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press Boston, MA [3] Golpalsamy, K., Globally asymptotic stability in a periodic integrodifferential system, Tohoku math. J., 3, 2^{nd} ed. Ser., 323-332, (1985) [4] Alvarez, C.; Lazer, A.C., An application of topological degree to the periodic competiting species model, J. austral. math. soc. ser. B, 28, 202-219, (1986) · Zbl 0625.92018 [5] Battaaz, A.; Zanolin, F., Coexistence states for periodic competitive Kolmogorov systems, J. math. anal. appl., 219, 179-199, (1998) · Zbl 0911.34037 [6] Korman, P., Some new results on the periodic competition model, J. math. anal. appl., 171, 131-138, (1992) · Zbl 0848.34026 [7] Smith, H.L., Periodic solutions of periodic competitive and cooperative systems, SIAM J. math. anal., 17, 1289-1318, (1986) · Zbl 0609.34048 [8] Trieo, A.; Alrarez, C., A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J. math. anal. appl., 159, 44-50, (1991) [9] Golpalsamy, K.; Weng, P., Feedback regulation of logistic growth, Internat. J. math. and math. sci., 16, 177-192, (1993) · Zbl 0765.34058 [10] Weng, P., Global attractivity in a competition system with feedback control and time delay, (), 383-388, Series 176 · Zbl 0844.34084 [11] Weng, P., Global attractivity in a periodic competition system with feedback controls, Acta appl. math., 12, 11-21, (1996) · Zbl 0859.34061 [12] Aizerman, M.A.; Gantmacher, F.R., Absolute stability of regulator systems, (1964), Holden Day San Francisco, CA, translated from Russian · Zbl 0123.28401 [13] Lefschetz, S., Stability of nonlinear control systems, (1965), Academic Press New York · Zbl 0136.08801 [14] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021 [15] Barbǎlat, I., Systemes d’equations differentielle d’oscillations nonlineaires, Rev. roumaine math. pures appl., 4, 267-270, (1959) · Zbl 0090.06601 [16] Ahmad, S., On the nonautonomous Volterra-Lotka competition equations, (), 199-204 · Zbl 0848.34033 [17] Freedman, H.I.; Waltman, P., Persistence in a model of three competitive populations, Math. biosci., 73, 89-101, (1985) · Zbl 0584.92018 [18] May, R.M.; Leonard, W.J., Nonlinear aspects of competition between three species, SIAM J. appl. math., 29, 243-253, (1975) · Zbl 0314.92008
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.