# zbMATH — the first resource for mathematics

Permanence of a nonlinear prey-competition model with delays. (English) Zbl 1131.92062
Summary: A nonlinear periodic predator-prey model with $$m$$-preys and $$(n - m)$$-predators and delays is studied, which can be seen as the modification of the traditional Lotka-Volterra prey-competition model. Two sets of sufficient conditions which guarantee the permanence of the system are obtained. One set is delay independent, while the other set is delay dependent. Our results supplement those obtained by F. D. Chen, X. Xie and J. Shi [Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays. J. Comput. Appl. Math. 194, No. 2, 368–387 (2006; Zbl 1104.34050)].

##### MSC:
 92D40 Ecology 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K20 Stability theory of functional-differential equations
nonautonomous
Full Text:
##### References:
 [1] Chen, F.D.; Xie, X.D.; Shi, J.L., Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, Journal of computational and applied mathematics, 194, 2, 368-387, (2006) · Zbl 1104.34050 [2] Chen, F.D., On a nonlinear non-autonomous predator – prey model with diffusion and distributed delay, Journal of computational and applied mathematics, 180, 1, 33-49, (2005) · Zbl 1061.92058 [3] Li, C.R.; Lu, S.J., The qualitative analysis of N-species periodic coefficient, nonlinear relation, prey-competition systems (in Chinese), Applied mathematics - JCU, 12, 2, 147-156, (1997) · Zbl 0880.34042 [4] Zhao, J.D.; Chen, W.C., The qualitative analysis of N-species nonlinear prey-competition systems, Applied mathematics and computation, 149, 2, 567-576, (2004) · Zbl 1045.92038 [5] Fan, M.; Wang, K., Global periodic solutions of a generalized n-species gilpin – ayala competition model, Computer and mathematics with applications, 40, 1141-1151, (2000) · Zbl 0954.92027 [6] Yang, P.; Xu, R., Global attractivity of the periodic lotka – volterra system, Journal of mathematical analysis and applications, 233, 1, 221-232, (1999) · Zbl 0973.92039 [7] Huo, H.F.; Li, W.T., Periodic solutions of a periodic lotka – volterra system with delay, Applied mathematics and computation, 156, 3, 787-803, (2004) · Zbl 1069.34099 [8] Yan, J.; Feng, Q., Global existence and oscillation in a nonlinear delay equation, Nonlinear analysis, 43, 101-108, (2001) · Zbl 0987.34065 [9] Chen, F.D.; Lin, S.J., Periodicity in a logistic type system with several delays, Computer and mathematics with applications, 48, 1-2, 35-44, (2004) · Zbl 1061.34050 [10] Li, Y.K., Periodic solutions for delay lotka – volterra competition systems, Journal of mathematical analysis and applications, 246, 1, 230-244, (2000) · Zbl 0972.34057 [11] Chen, F.D., On a periodic multi-species ecological model, Applied mathematics and computation, 171, 1, 492-510, (2005) · Zbl 1080.92059 [12] Chen, F.D., Average conditions for permanence and extinction in nonautonomous gilpin – ayala competition model, Nonlinear analysis: real world applications, 7, 4, 895-915, (2006) · Zbl 1119.34038 [13] Chen, F.D., Some new results on the permanence and extinction of nonautonomous gilpin – ayala type competition model with delays, Nonlinear analysis: real world applications, 7, 5, 1205-1222, (2006) · Zbl 1120.34062 [14] Chen, F.D.; Shi, C.L., Global attractivity in an almost periodic multi-species nonlinear ecological model, Applied mathematics and computation, 180, 1, 376-392, (2006) · Zbl 1099.92069
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.