# zbMATH — the first resource for mathematics

Permanence of a delayed discrete mutualism model with feedback controls. (English) Zbl 1185.93050
Summary: This paper discusses a two species discrete model of mutualism with delays and feedback controls. Sufficient conditions are obtained for the permanence of the system. The results show that feedback control variables have no influence on the persistence property of the system.

##### MSC:
 93B52 Feedback control 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Janzen, D.H., Coevolution of mutualism between ants and acacias in central America, Evolution, 20, 249-275, (1966) [2] Porter, K.G., Enhancement of algal growth and productivity by grazing zooplankton, Science, 192, 1332-1334, (1976) [3] Goh, B.S., Stability in models of mutualism, Amer. nat., 113, 216-275, (1979) [4] Zhang, Z.B., Mutualism or cooperation among competitors promotes coexistence and competitive ability, Ecol. modelling, 164, 271-282, (2003) [5] Chen, L.S.; Lu, Z.Y.; Wang, W.D., The effect of delays on the permanence for lotka – volterra systems, Appl. math. lett., 8, 4, 71-73, (1995) · Zbl 0833.34071 [6] Gapalsamy, K., Stability and oscillations in delay equations of population dynamics, (1992), Kluwer Academic Publishers London, pp. 191-192 [7] Li, Y.K.; Xu, G.T., Positive periodic solutions for an integrodifferential model of mutualism, Appl. math. lett., 14, 5, 525-530, (2001) · Zbl 0981.45002 [8] Chen, F.D.; You, M.S., Permanence for an integrodifferential model of mutualism, Appl. math. comput., 186, 1, 30-34, (2007) · Zbl 1118.45006 [9] Vandermeer, J.H.; Boucher, D.H., Varieties of mutualistic interaction in population models, J. theoret. biol., 74, 549-558, (1978) [10] Boucher, D.H.; James, S.; Keeler, K.H., The ecology of mutualism, Ann. rev. ecol. syst., 13, 315-347, (1982) [11] Dean, A.M., A simple model of mutualism, Amer. natural., 121, 409-417, (1983) [12] Wolin, C.L.; Lawlor, L.R., Models of facultative mutualism: density effects, Amer. natural., 144, 843-862, (1984) [13] Boucher, D.H., The biology of mutualism: ecology and evolution, (1985), Croom Helm London [14] Li, Y.K., Positive periodic solutions of a discrete mutualism model with time delays, Int. J. math. math. sci., 2005, 4, 499-506, (2005) · Zbl 1081.92042 [15] Chen, F.D., Permanence for the discrete mutualism model with time delays, Math. comput. modelling, 47, 431-435, (2008) · Zbl 1148.39017 [16] Cui, J.A., Global asymptotic stability in $$n$$-species cooperative system with time delays, Systems sci. math. sci., 7, 1, 45-48, (1994) · Zbl 0807.34088 [17] Yang, P.; Xu, R., Global asymptotic stability of periodic solution in $$n$$-species cooperative system with time delays, J. biomath., 13, 6, 841-846, (1998) [18] Wei, F.Y.; Wang, K., Asymptotically periodic solution of $$N$$-species cooperation system with time delay, Nonlinear anal. RWA, 7, 4, 591-596, (2006) · Zbl 1114.34340 [19] Chen, F.D., Permanence of a discrete $$N$$-species cooperation system with time delays and feedback controls, Appl. math. comput., 186, 1, 23-29, (2007) · Zbl 1113.93063 [20] Chen, F.D.; Liao, X.Y.; Huang, Z.K., The dynamic behavior of $$N$$-species cooperation system with continuous time delays and feedback controls, Appl. math. comput., 181, 2, 803-815, (2006) · Zbl 1102.93021 [21] 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 [22] Chen, F.D., Permanence of a single species discrete model with feedback control and delay, Appl. math. lett., 20, 729-733, (2007) · Zbl 1128.92029 [23] Fan, Y.H.; Wang, Lin-lin, Permanence for a discrete model with feedback control and delay, Discrete dyn. nat. soc., 2008, 8 pp, (2008), Article ID 945109 · Zbl 1149.39003 [24] Li, Y.K.; Zhu, L.F., Existence of a positive periodic solutions for difference equations with feedback control, Appl. math. lett., 18, 61-67, (2005) · Zbl 1085.39009 [25] Li, Y.K., Positive periodic solutions for a periodic neutral differential equation with feedback control, Nonlinear anal. RWA, 6, 1, 145-154, (2005) · Zbl 1092.34033 [26] 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 [27] Chen, F.D., The permanence and global attractivity of lotka – voterra competition system with feedback controls, Nonlinear anal. RWA, 7, 1, 133-143, (2006) · Zbl 1103.34038 [28] Yang, X.T., Uniform persistence and periodic solutions for a discrete predator – prey system with delays, J. math. anal. appl., 316, 161-177, (2006) · Zbl 1107.39017
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.