×

Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response. (English) Zbl 1175.34058

The authors consider a Volterra model with mutual interference and Beddington-DeAngelis functional response. By applying the comparison theorem for differential equations and constructing a suitable Lyapunov functional, sufficient conditions for permanence and existence of a unique globally attractive positive almost periodic solution are obtained. A suitable example together with its numeric simulations is given to illustrate the feasibility of the main results.

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wang, K.; Zhu, Y., Global attractivity of positive periodic solution for a Volterra model, Appl. Math. Comput., 203, 2, 493-501 (2008) · Zbl 1178.34052
[2] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[3] Chen, F. D., Periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sinica Engl. Ser., 21, 1, 1-10 (2005)
[4] Shen, C. X., Permanence and global attractivity of the food-chin system with Holling IV type functional response, Appl. Math. Comput., 194, 1, 179-185 (2007) · Zbl 1193.34142
[5] Zhang, L.; Teng, Z. D., Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent, J. Math. Anal. Appl., 338, 1, 175-193 (2008) · Zbl 1147.34056
[6] Chen, F.; Cao, X. H., Existence of almost periodic solution in a ratio-dependent Leslie system with feedback controls, J. Math. Anal. Appl., 341, 1399-1412 (2008) · Zbl 1145.34026
[7] Song, X. Y.; Li, Y. F., Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonlinear Anal. Real World Appl., 9, 1, 64-79 (2008) · Zbl 1142.34031
[8] Zhou, X. Y.; Shi, X. Y.; Song, X. Y., Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, Appl. Math. Comput., 196, 129-136 (2008) · Zbl 1147.34355
[9] Meng, X. Z.; Xu, W. J.; Chen, L. S., Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion, Appl. Math. Comput., 188, 1, 365-378 (2007) · Zbl 1113.92070
[10] Liu, S. Q.; Zhang, J. H., Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure, J. Math. Anal. Appl., 342, 1, 446-460 (2008) · Zbl 1146.34057
[11] Ding, X. Q.; Jiang, J. F., Multiple periodic solutions in delayed Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses, Math. Comput. Modell., 47, 11, 1323-1331 (2008) · Zbl 1145.34332
[12] Hassel, M. P., Density dependence in single-species population, J. Anim. Ecol., 44, 283-295 (1975)
[13] Chen, L. S., Mathematics Ecology Models and Research Methods (1988), Science Press: Science Press Beijing, (in Chinese)
[14] Fan, M.; Kuang, Y., Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 259, 15-39 (2004) · Zbl 1051.34033
[15] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139, 1287-1296 (1989)
[16] Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551 (1992)
[17] Gutierrez, A. P., The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholsons blowflies as an example, Ecology, 73, 1552-1563 (1992)
[18] Zhou, Z. H.; Yang, Z. H., Periodic solutions in higher-dimensional Lotka-Volterra neutral competition systems with state-dependent delays, Appl. Math. Comput., 189, 1, 986-995 (2007) · Zbl 1134.34046
[19] Han, F.; Wang, Q. Y., Existence of multiple positive periodic solutions for differential equation with state-dependent delays, J. Math. Anal. Appl., 324, 2, 908-920 (2006) · Zbl 1112.34049
[20] Chen, F. D., Almost periodic solution of the non-autonomous two-species competitive model with stage structure, Appl. Math. Comput., 181, 685-693 (2006) · Zbl 1163.34030
[21] Shi, C. L.; Li, Z.; Chen, F. D., The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays, Nonlinear Anal. Real World Appl., 8, 5, 536-1550 (2007) · Zbl 1128.92053
[22] Feng, C. H.; Liu, Y. J., Almost periodic solutions for delay Lotka-Volterra competitive systems, Acta Math. Appl. Sinica, 28, 3, 459-465 (2005), (in Chinese)
[23] Feng, C. H.; Wang, P. G., The existence of almost periodic solutions of some delay differential equations, Comput. Math. Appl., 47, 8, 1225-1231 (2004) · Zbl 1080.34561
[24] Yang, X. T.; Yuan, R., Global attractivity and positive almost periodic solution for delay logistic differential equation, Nonlinear Anal., 68, 1, 54-72 (2008) · Zbl 1136.34057
[25] Yang, X. T., Global attractivity and positive almost periodic solution of a single species population model, J. Math. Anal. Appl., 336, 1, 111-126 (2007) · Zbl 1132.34050
[26] Liu, S. Q.; Chen, L. S., Permanence, extinction and balancing survival in nonautonomous system with delays, Appl. Math. Comput., 129, 481-499 (2002) · Zbl 1035.34088
[27] He, C. Y., Almost Periodic Differential Equations (1992), Higher Education Press, in Chinese
[28] A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin, 1974.; A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin, 1974. · Zbl 0325.34039
[29] Wang, Q.; Dai, B. X., Almost periodic solution for \(n\)-species Lotka-Volterra competitive system with delay and feedback controls, Appl. Math. Comput., 200, 1, 133-146 (2008) · Zbl 1146.93021
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.