zbMATH — the first resource for mathematics

Global dynamics of Nicholson-type delay systems with applications. (English) Zbl 1208.34120
Summary: Models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics that belong to the Nicholson-type delay differential systems are proposed. To study the global stability of the Nicholson-type models, we construct an exponentially stable linear system such that its solution is a solution of the nonlinear model. Explicit conditions of the existence of positive global solutions, lower and upper estimations of solutions, and the existence and uniqueness of a positive equilibrium are obtained. New results, obtained for the global stability and instability of equilibria solutions, extend known results for the scalar Nicholson models. The conditions for the stability test are quite practical, and the methods developed are applicable to the modeling of a broad spectrum of biological processes. To illustrate our finding, we study the dynamics of the fish populations in marine protected areas.

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
Full Text: DOI
[1] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (2001), Springer-Verlag · Zbl 0967.92015
[2] Kuang, Y., ()
[3] Sumalia, U., Addressing ecosystem effects of fishing using marine protected areas, ICES J. mar. sci., 57, 752-760, (2000)
[4] Araujo, R.; McElwain, D., A history of the study of solid tumour growth: the contribution of mathematical modelling, Bull. math. biol., 66, 5, 1039-1091, (2004) · Zbl 1334.92187
[5] Afenya, E., Recovery of normal hemopoiesis in disseminated cancer therapy—a model, Math. biosci., 172, 15-32, (2001) · Zbl 1003.92018
[6] Kozusko, F.; Bajzer, Z., Combining Gompertzian growth and cell population dynamics, Math. biosci., 185, 153-167, (2003) · Zbl 1021.92012
[7] Menasria, F., Apoptotic effects on B-cell chronic lymphocytic leukemia (B-CLL) cells of heterocyclic compounds isolated from guttiferaes, Leuk. res., 32, 1914-1926, (2008)
[8] Messmer, B., In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells, J. clin. invest., 115, 755-764, (2005)
[9] Berezansky, L.; Braverman, E.; Idels, L., Nicholsons blowflies differential equations revisited: main results and open problems, Appl. math. model., 34, 6, 1405-1417, (2010) · Zbl 1193.34149
[10] Gil’, M., ()
[11] Gopalsamy, K., ()
[12] Davydov, V.; Khusainov, D., Stability investigation of quadratic systems with delay, J. appl. math. stoch. anal., 13, 85-92, (2000) · Zbl 0956.34064
[13] Diblìk, J.; Koksch, N., Sufficient conditions for the existence of global solutions of delayed differential equations, J. math. anal. appl., 318, 611-625, (2006) · Zbl 1102.34048
[14] Diblìk, J., Existence of bounded solutions of retarded functional differential equations, Math. nachr., 226, 49-64, (2001) · Zbl 0985.34060
[15] Faria, T., Sharp conditions for global stability of lotka – volterra systems with distributed delays, J. differential equations, 246, 4391-4404, (2009) · Zbl 1172.34051
[16] Gil’, M., \(L^2\)-stability of vector equations with nonlinear causal mappings, Dynam. systems appl., 17, 201-219, (2008) · Zbl 1162.34059
[17] Gil’, M., Explicit stability conditions for a class of semilinear retarded systems, Internat. J. control, 80, 322-327, (2007) · Zbl 1133.93360
[18] Gyòri, I.; Hartung, F., Exponential stability of a state-dependent delay system, Discrete contin. dyn. syst., 18, 773-791, (2007) · Zbl 1144.34051
[19] Gyòri, I.; Hartung, F., Fundamental solution and asymptotic stability of linear delay differential equations, Dyn. contin. discrete impuls. syst. ser. A math. anal., 13, 261-287, (2006) · Zbl 1099.34067
[20] Krisztin, T., Global dynamics of delay differential equations, Period. math. hungar., 56, 83-95, (2008) · Zbl 1164.34037
[21] Miguel, J.; Ponosov, A.; Shindiapin, A., On a delay equation with richards’ nonlinearity, (), 3919-3924 · Zbl 1042.34587
[22] Muroya, Y., Global stability for separable nonlinear delay differential systems, J. math. anal. appl., 326, 372-389, (2007) · Zbl 1186.34098
[23] Gyòri, I.; Hartung, F., Stability results for cellular neural networks with delays, Proceedings of the 7th colloquium on the qualitative theory of differential equations, J. qual. theory differ. equ., 13, 1-14, (2004)
[24] Idels, L.; Kipnis, M., Stability criteria for a nonautonomous nonlinear system with delay, Appl. math. model., 33, 2293-2297, (2008) · Zbl 1185.74043
[25] Liu, B., Global stability of a class of delay differential systems, J. comput. appl. math., 233, 217-223, (2009) · Zbl 1189.34145
[26] Liu, S.; Chen, L.; Agarwal, R., Recent progress on stage-structured population dynamics, Math. comput. modelling, 36, 1319-1360, (2002) · Zbl 1077.92516
[27] Liz, E.; Pinto, M.; Tkachenko, V.; Trofimchuk, S., A global stability criterion for a family of delayed population models, Quart. appl. math., 63, 56-70, (2005) · Zbl 1093.34038
[28] Mohamad, S.; Gopalsamy, K.; Akca, H., Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear anal. RWA, 9, 872-888, (2008) · Zbl 1154.34042
[29] Xia, Y.; Wong, P., Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos solitons fractals, 39, 440-453, (2009) · Zbl 1197.34146
[30] Hale, J.; Lunel, S., ()
[31] Lozinskiǐ, S., Error estimate for numerical integration of ordinary differential equations, I, Izv. vyssh. uchebn. zaved. mat., 5, 52-90, (1958), (in Russian) · Zbl 0198.21202
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.