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Global attractivity in an almost periodic multi-species nonlinear ecological model. (English) Zbl 1099.92069
Summary: A nonlinear almost periodic predator-prey model with \(n\)-preys and \(m\)-predators is studied, which can be seen as the modification of the traditional multi-species Lotka-Volterra predator-prey model. For the general nonautonomous case, by using differential inequality theory, we obtain the sufficient conditions which guarantee the uniform persistence and nonpersistence of the system. After that, by constructing a suitable Lyapunov function, some sufficient conditions are obtained which ensure the global attractivity of the system. For the almost periodic case, by constructing a suitable Lyapunov function, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained. Examples together with their numeric simulations show the feasibility of our main results.

92D40 Ecology
34A40 Differential inequalities involving functions of a single real variable
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI
[1] Gopalsamy, K., Global asymptotic stability in an almost periodic Lotka-Volterra system, J. austral. math. soc. ser. B, 27, 346-360, (1986) · Zbl 0591.92022
[2] Ahmad, S., On almost periodic solutions of the competing species problems, Proc. amer. math., 102, 855-865, (1988) · Zbl 0668.34042
[3] Chen, F.D.; Chen, X.X., n-competing Lotka-Volterra almost periodic systems with grazing rates, J. biomath., 18, 4, 411-416, (2003), (in Chinese)
[4] Tuan Anh, Trinh, On the almost periodic n-competing species problem, Acta math. Vietnam., 23, 1, 35-48, (1998) · Zbl 0916.34048
[5] Teng, Z.D., On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays, J. math. anal. appl., 249, 2, 433-444, (2000) · Zbl 0967.34064
[6] Chattopadhyay, J., Effect of toxic substance on a two-species competitive system, Ecol. model, 84, 287-289, (1996)
[7] Cui, J.; Chen, L., Asymptotic behavior of the solution for a class of time-dependent competitive system with feedback controls, Ann. differ. eqs., 9, 1, 11-17, (1993)
[8] Maynard-Smith, J., Models in ecology, (1974), Cambridge University Cambridge · Zbl 0312.92001
[9] Ayala, F.J.; Gilpin, M.E.; Eherenfeld, J.G., Competition between species: theoretical models and experimental tests, Theor. population biol., 4, 331-356, (1973)
[10] Gilpin, M.E.; Ayala, F.J., Global models of growth and competition, Proc. natl. acad. sci. USA, 70, 3590-3593, (1973) · Zbl 0272.92016
[11] Fan, M.; Wang, K., Global periodic solutions of a generalized n-species gilpin-ayala competition model, Comput. math. appl., 40, 1141-1151, (2000) · Zbl 0954.92027
[12] Zhang, S.W.; Tan, D.J.; Chen, L.S., The periodic n-species gilpin-ayala competition system with impulsive effect, Chaos, solitons & fractals, 26, 2, 507-517, (2005) · Zbl 1065.92065
[13] Berryman, A.A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535, (1992)
[14] Yang, P.; Xu, R., Global attractivity of the periodic Lotka-Volterra system, J. math. anal. appl., 233, 1, 221-232, (1999) · Zbl 0973.92039
[15] Zhao, J.D.; Chen, W.C., Global asymptotic stability of a periodic ecological model, Appl. math. comput., 147, 3, 881-892, (2004) · Zbl 1029.92026
[16] Xia, Y.; Chen, F.; Cao, J.; Chen, A., Existence and global attractivity of an almost periodic ecological model, Appl. math. comput., 157, 2, 449-475, (2004) · Zbl 1049.92038
[17] Li, C.R.; Lu, S.J., The qualitative analysis of N-species periodic coefficient, nonlinear relation, prey-competition systems, Appl. math. JCU, 12, 2, 147-156, (1997), (in Chinese) · Zbl 0880.34042
[18] Zhao, J.D.; Chen, W.C., The qualitative analysis of N-species nonlinear prey-competition systems, Appl. math. comput., 149, 2, 567-576, (2004) · Zbl 1045.92038
[19] Chen, F.D., On a periodic multi-species ecological model, Appl. math. comput., 171, 1, 492-510, (2005) · Zbl 1080.92059
[20] Chen, F.D.; Shi, J.L., Periodicity in a logistic type system with several delays, Comput. math. appl., 48, 1-2, 35-44, (2004) · Zbl 1061.34050
[21] Chen, F.D.; Sun, D.X.; Shi, J.L., Periodicity in a food-limited population model with toxicants and state dependent delays, J. math. anal. appl., 288, 1, 132-142, (2003)
[22] Chen, F.D., Periodicity in a nonlinear predator-prey system with state dependent delays, Acta math. appl. sinica, English series, 21, 1, 49-60, (2005) · Zbl 1096.34050
[23] Chen, F.D.; Lin, F.X.; Chen, X.X., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. math. comput., 158, 1, 45-68, (2004) · Zbl 1096.93017
[24] 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
[25] Chen, F.D., On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. comput. appl. math., 180, 1, 33-49, (2005) · Zbl 1061.92058
[26] Zhang, H.Y.; Chen, F.D.; Chen, X.X., Permanence and almost periodic solution for non-autonomous ratio-dependent multi-species competition predator-prey system, Afr. diaspora J. math., 2, 1, 1-12, (2004) · Zbl 1106.34033
[27] F.D. Chen, X.D. Xie, J.L. Shi, Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math., in press. · Zbl 1104.34050
[28] Fink, A.M., Almost periodic differential equations, Lecture notes in math, vol. 377, (1974), Springer-Verlag Berlin · Zbl 0325.34039
[29] He, C.Y., Almost periodic differential equations, (1992), Higher Education Press, (in Chinese)
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