Almost periodic solution of the non-autonomous two-species competitive model with stage structure. (English) Zbl 1163.34030

The system of differential equations
\[ \begin{cases} \dot{x}_1(t)=-a_1(t)x_1(t)+b_1(t)x_2(t),\\ \dot{x}_2(t)=a_2(t)x_2(t)-b_2(t)x_2(t)-c(t)x^2_2(t)-\beta_1(t)x_2(t)x_3(t),\\ \dot{x}_3(t)=x_3(t)(d(t)-e(t)x_3(t)-\beta_2(t)x_2(t)), \end{cases} \]
as a non-autonomous two-species competitive model with stage structure is investigated. The aim of this article is, by further developing the analytic technique of K. Gopalsamy [J. Aust. Math. Soc., Ser. B 27, 346–360 (1986; Zbl 0591.92022)] and by constructing a suitable Lyapunov function, to obtain a set of “easily verifiable” sufficient conditions to ensure the existence of a unique, globally attractive, positive , almost periodic solution of system (1). An example is presented.


34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations


Zbl 0591.92022
Full Text: DOI


[1] Zeng, G. Z.; Chen, L. S.; Sun, L. H.; Liu, Y., Permanence and the existence of the periodic solution of the non-autonomous two-species competitive model with stage structure, Adv. Complex Syst., 7, 3&4, 385-393 (2004) · Zbl 1080.34037
[2] Chen, L. S.; Chen, J., Nonlinear Biological Dynamics Systems (1993), Science Press: Science Press Beijing, [in Chinese]
[3] Cui, J. A.; Chen, L. S., The effect of dispersal on population growth with stage-structure, Comput. Math. Appl., 39, 1&2, 91-102 (2000) · Zbl 0968.92018
[4] Liu, S. Q.; Chen, L. S., Recent progress on stage-structure population dynamics, Math. Comput. Modell., 36, 1319-1360 (2002) · Zbl 1077.92516
[5] Song, X. Y.; Chen, L. S., Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170, 173-186 (2000)
[6] Xiao, Y. N.; Chen, L. S., Stabilizing effect of cannibalism on a structured competitive system, Acta Math. Sci., Ser. A, 22, 2, 210-216 (2002), [in Chinese] · Zbl 1041.34033
[7] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator-prey model, J. Math. Anal. Appl., 262, 2, 499-528 (2001) · Zbl 0997.34069
[8] Wang, W.; Chen, L., A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33, 8, 83-91 (1997)
[9] Zhang, Z. Q.; Wu, J.; Wang, Z. C., Periodic solutions of nonautonomous stage-structured cooperative system, Comput. Math. Appl., 47, 4-5, 699-706 (2004) · Zbl 1069.34066
[10] Chen, F. D., Periodicity in a ratio-dependent predator-prey system with stage structure for predator, J. Appl. Math., 2005, 2, 153-169 (2005) · Zbl 1103.34060
[11] 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
[12] Gopalsamy, K., Global asymptotic stability in an almost periodic Lotka-Volterra system, J. Aust. Math. Soc. Ser. B, 27, 346-360 (1986) · Zbl 0591.92022
[13] He, C. Y., Almost Periodic Differential Equations (1992), Higher Education Press, [in Chinese]
[14] Fink, A. M., Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377 (1974), Springer-Verlag: Springer-Verlag Berlin · Zbl 0325.34039
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.