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

Weng, Peixuan

Comput. Math. Appl. 40, No. 6-7, 747-759 (2000).

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.

Reviewer: Nikolay Yakovlevich Tikhonenko (Odessa)

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

Keywords:

population dynamics; periodic competition model; feedback controls; integro-differential system; delay differential equations; indirect control variables; positive periodic solution; coincidence degree theory; Lyapunov functional; global asymptotic stability

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