## $$N$$-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations.(English)Zbl 1231.37055

Summary: A class of non-autonomous $$N$$-species Lotka-Volterra-type competitive system with time delays and impulsive perturbations is investigated. New criteria on coexistence and global attractivity for all species other than some inclining to extinction are established. From our results, we find that the extinction of some species due to competition exclusion and weak adaption to a stochastically variable environment characterized by impulsive perturbations is necessary for the coexistence and global attractivity of other survivors in a limited resource ecosystem. Meanwhile, the dynamic behaviors of competition models with impulse perturbations are more complicated and different from prior research results obtained from continuous competition models. The impulse perturbations impose either negative or positive influences on the survival of species, which results in evidently regulative effects, and even make the inferior competitors exclude or coexist with the superior competitors. The theoretical results are confirmed by a special example and numerical simulations, by which we find some interesting phenomena.

### MSC:

 37N25 Dynamical systems in biology 92D25 Population dynamics (general)
Full Text:

### References:

 [1] Miller, R.S., Pattern and process in competition, (), 1-74 [2] Levin, S., Community equilibria and stability, and an extension of the competitive exclusion principle, Am. nat., 104, 413-423, (1970) [3] Terborgh, J., On the notion of favorableness in plant ecology, Am. nat., 107, 481-501, (1973) [4] MacArthur, R.H., Patterns of species diversity, Biol. rev., 40, 510-533, (1965) [5] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge University Press New York [6] Grinnell, J., The origin and distribution of the chestnut-backed chickadee, Auk, 21, 364-379, (1904) [7] Gause, G.F., The struggle for existence, (1934), Hafner Press New York, NY, Williams and Wilkins, Baltimore, MD. Reprinted in 1964 by Macmillan [8] Hardin, G., The competitive exclusion principle, Science, 131, 1292-1297, (1960) [9] Lotka, J.A., Elements of physical biology, (1925), Williams & Wilkins Baltimore, p. 460 · JFM 51.0416.06 [10] Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature (London), 118, 558-560, (1926) · JFM 52.0453.03 [11] Freedman, H.I.; Waltman, P., Persistence in models of three competitive populations, Math. biosci., 73, 89-101, (1985) · Zbl 0584.92018 [12] Smith, H.; Zhao, X., Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete contin. dyn. syst. ser. B, 1, 183-191, (2001) · Zbl 1056.92059 [13] Tineo, A., An iterative scheme for the $$N$$-competing species problem, J. differential equations, 116, 1-15, (1995) · Zbl 0823.34048 [14] Zanolin, F., Permanence and positive periodic solutions for Kolmogorov competing species systems, Results math., 21, 224-250, (1992) · Zbl 0765.92022 [15] Zeeman, M., Extinction in competitive lotka – volterra systems, Proc. amer. math. soc., 123, 87-96, (1995) · Zbl 0815.34039 [16] de Oca, F. Montes; Zeeman, M.L., Extinction in nonautonomous competitive lotka – volterra systems, Proc. amer. math. soc., 124, 3677-3687, (1996) · Zbl 0866.34029 [17] Ahmad, S., Extinction of species in nonautonomous lotka – volterra systems, Proc. amer. math. soc., 127, 2905-2910, (1999) · Zbl 0924.34040 [18] Ahmad, S.; Stamova, I.M., Partial persistence and extinction in $$N$$-dimensional competitive systems, Nonlinear anal. TMA, 60, 821-836, (2005) · Zbl 1071.34046 [19] Ahmad, S.; Mohana, R., Asymptotically periodic solutions of $$n$$-competing species problem with time delay, J. math. anal. appl., 186, 557-571, (1994) [20] Ahmad, S.; Granados, B.; Tineo, A., On tridiagonal predator – prey systems and a conjecture, Nonlinear anal. RWA, 11, 3, 1878-1881, (2010) · Zbl 1200.34059 [21] Teng, Z.; Yu, Y., Some new results of nonautonomous lotka – volterra competitive systems with delays, J. math. anal. appl., 241, 254-275, (2000) · Zbl 0947.34066 [22] Li, Y.; Zhao, K.; Ye, Y., Multiple positive periodic solutions of $$n$$ species delay competition systems with harvesting terms, Nonlinear anal. RWA, 12, 2, 1013-1022, (2011) · Zbl 1225.34094 [23] Zhang, L.; Teng, Z., Permanence for a class of periodic time-dependent competitive system with delays and dispersal in a patchy-environment, Appl. math. comput., 188, 855-864, (2007) · Zbl 1124.34058 [24] Elton, C.S., The ecology of invasions by animals and plants, (1958), Methuen London [25] May, R.M., Stability and complexity in model ecosystems, (1974), Princeton University Press Princeton, NJ [26] Tilman, D.; Lehman, C.L.; Bristow, C.E., Diversity – stability relationship: statistical inevitability or ecological consequence?, Am. nat., 151, 277-282, (1998) [27] Bartlett, M.S., Stochastic population models, (1960), Methuen and Co. London · Zbl 0096.13702 [28] May, R.M., Time-delay versus stability in population models with two and three trophic levels, Ecology, 54, 2, 315-325, (1973) [29] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [30] Bainov, D.; Simeonov, P., Impulsive differential equations: periodic solutions and applications, (2003), Longman England · Zbl 1085.34557 [31] Gopalsamy, K.; Zhang, B.G., On delay differential equations with impulses, J. math. anal. appl., 139, 110-122, (1989) · Zbl 0687.34065 [32] Jin, Z.; Han, M.; Li, G., The persistence in a lotka – volterra competition systems with impulsive, Chaos solitons fractals, 24, 1105-1117, (2005) · Zbl 1081.34045 [33] Zhang, L.; Teng, Z.; Jiang, H., Permanence for general nonautonomous impulsive population systems of functional differential equations and its applications, Acta appl. math., 110, 1169-1197, (2010) · Zbl 1186.92052 [34] Hou, J.; Teng, Z.; Gao, S., Permanence and global stability for nonautonomous $$N$$-species lotka – volterra competitive system with impulses, Nonlinear anal. RWA, 11, 1882-1896, (2010) · Zbl 1200.34051 [35] Hu, H.; Wang, K.; Wu, D., Permanence and global stability for nonautonomous $$N$$-species lotka – volterra competitive system with impulses and infinite delays, J. math. anal. appl., 377, 1, 145-160, (2011) · Zbl 1223.34112 [36] He, M.; Li, Z.; Chen, F., Permanence, extinction and global attractivity of the periodic gilpin – ayala competition system with impulses, Nonlinear anal. RWA, 11, 1537-1551, (2010) · Zbl 1255.34056 [37] Wang, W.; Shen, J.; Luo, Z., Partial survival and extinction in two competing species with impulses, Nonlinear anal. RWA, 10, 1243-1254, (2009) · Zbl 1162.34308 [38] Ahmad, S.; Stamova, I.M., Asymptotic stability of an $$N$$-dimensional impulsive competitive system, Nonlinear anal. RWA, 8, 654-663, (2007) · Zbl 1152.34342 [39] Ahmad, S.; Stamova, I.M., Asymptotic stability of competitive systems with delays and impulsive perturbations, J. math. anal. appl., 334, 686-700, (2007) · Zbl 1153.34044 [40] Ahmad, S.; Stamov, G.T., On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations, Nonlinear anal. RWA, 10, 5, 2857-2863, (2009) · Zbl 1170.45004 [41] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. RWA, 10, 2, 680-690, (2009) · Zbl 1167.34318 [42] Stamova, I.M., Impulsive control for stability of $$n$$-species lotka – volterra cooperation models with finite delays, Appl. math. lett., 23, 9, 1003-1007, (2010) · Zbl 1200.34103 [43] Yan, J.; Zhao, A.; Nieto, J.J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive lotka – volterra systems, Math. comput. modelling, 40, 5-6, 509-518, (2004) · Zbl 1112.34052 [44] Tang, S.Y.; Cheke, R.A., State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. math. biol., 50, 257-292, (2005) · Zbl 1080.92067 [45] Liu, X.; Chen, L., Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. math. anal. appl., 289, 279-291, (2004) · Zbl 1054.34015 [46] Teng, Z.; Li, Z., Permanence and asymptotic behavior of the $$N$$-species nonautonomous lotka – volterra competitive systems, Comput. math. appl., 39, 107-116, (2000) · Zbl 0959.34039 [47] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. biosci., 111, 1-71, (1992) · Zbl 0783.92002 [48] Thieme, H., Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. biosci., 166, 173-201, (2000) · Zbl 0970.37061
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.